A current-sensorless PWM-based robust sliding mode controller is proposed for the DC-DC Boost Converter, a nonminimum phase system that presents major challenges in the design of stabilizing controllers. The development of the controller requires the measurement of the output voltage and the estimation of its derivative. An extended state observer is developed to estimate a lumped uncertainty that comprises the uncertain load and input voltage, the converter parasitics, and the component uncertainties, and also to estimate the derivative of the output voltage. A linear sliding surface is used to derive the controller that is simple in its design and yet exhibits excellent features in terms of robustness to external disturbances, parameter uncertainties, and parasitics, despite the absence of the inductor current feedback. Also, a simple procedure to select the controller gains is outlined. The robustness of the controller is validated by computer simulations.
Citation: Said Oucheriah, Abul Azad. Current-sensorless robust sliding mode control for the DC-DC boost converter[J]. AIMS Electronics and Electrical Engineering, 2025, 9(1): 46-59. doi: 10.3934/electreng.2025003
[1] | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori . On generalized Hermite polynomials. AIMS Mathematics, 2024, 9(11): 32463-32490. doi: 10.3934/math.20241556 |
[2] | Mohra Zayed, Shahid Wani . Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials. AIMS Mathematics, 2023, 8(12): 30813-30826. doi: 10.3934/math.20231575 |
[3] | Mohra Zayed, Taghreed Alqurashi, Shahid Ahmad Wani, Cheon Seoung Ryoo, William Ramírez . Several characterizations of bivariate quantum-Hermite-Appell Polynomials and the structure of their zeros. AIMS Mathematics, 2025, 10(5): 11184-11207. doi: 10.3934/math.2025507 |
[4] | Gyung Won Hwang, Cheon Seoung Ryoo, Jung Yoog Kang . Some properties for 2-variable modified partially degenerate Hermite (MPDH) polynomials derived from differential equations and their zeros distributions. AIMS Mathematics, 2023, 8(12): 30591-30609. doi: 10.3934/math.20231564 |
[5] | Mohra Zayed, Shahid Ahmad Wani . Properties and applications of generalized 1-parameter 3-variable Hermite-based Appell polynomials. AIMS Mathematics, 2024, 9(9): 25145-25165. doi: 10.3934/math.20241226 |
[6] | Yunbo Tian, Sheng Chen . Prime decomposition of quadratic matrix polynomials. AIMS Mathematics, 2021, 6(9): 9911-9918. doi: 10.3934/math.2021576 |
[7] | Mohra Zayed, Shahid Ahmad Wani, William Ramírez, Clemente Cesarano . Advancements in q-Hermite-Appell polynomials: a three-dimensional exploration. AIMS Mathematics, 2024, 9(10): 26799-26824. doi: 10.3934/math.20241303 |
[8] | Mdi Begum Jeelani . On employing linear algebra approach to hybrid Sheffer polynomials. AIMS Mathematics, 2023, 8(1): 1871-1888. doi: 10.3934/math.2023096 |
[9] | Young Joon Ahn . An approximation method for convolution curves of regular curves and ellipses. AIMS Mathematics, 2024, 9(12): 34606-34617. doi: 10.3934/math.20241648 |
[10] | Nusrat Raza, Mohammed Fadel, Kottakkaran Sooppy Nisar, M. Zakarya . On 2-variable q-Hermite polynomials. AIMS Mathematics, 2021, 6(8): 8705-8727. doi: 10.3934/math.2021506 |
A current-sensorless PWM-based robust sliding mode controller is proposed for the DC-DC Boost Converter, a nonminimum phase system that presents major challenges in the design of stabilizing controllers. The development of the controller requires the measurement of the output voltage and the estimation of its derivative. An extended state observer is developed to estimate a lumped uncertainty that comprises the uncertain load and input voltage, the converter parasitics, and the component uncertainties, and also to estimate the derivative of the output voltage. A linear sliding surface is used to derive the controller that is simple in its design and yet exhibits excellent features in terms of robustness to external disturbances, parameter uncertainties, and parasitics, despite the absence of the inductor current feedback. Also, a simple procedure to select the controller gains is outlined. The robustness of the controller is validated by computer simulations.
In this paper, we are concerned with the sharp decay rates of solutions to the Cauchy problem for the isentropic Navier-Stokes equations:
{∂tρ+div(ρu)=0,(t,x)∈R+×R3,∂t(ρu)+div(ρu⊗u)+∇p(ρ)=divT,(t,x)∈R+×R3,lim|x|→∞ρ=ˉρ,lim|x|→∞u=0,t∈R+,(ρ,u)|t=0=(ρ0,u0),x∈R3, | (1.1) |
which governs the motion of a isentropic compressible viscous fluid. The unknown functions
Using the classical spectral method, the optimal time decay rate (upper bound) of the linearized equations of the isentropic Navier-Stokes equations are well known. One may then expect that the small solution of the nonlinear equations (1.1) have the same decay rate as the linear one. Our work is devoted to proving the sharp time decay rate (for both upper and lower bound) for the nonlinear system.
In the case of one space dimension, Zeng [24] and Liu-Zeng [15] offered a detailed analysis of the solution to a class of hyperbolic-parabolic system through point-wise estimate, including the isentropic Navier-Stokes system. For multi-dimensional Navier-Stokes equations (and/or Navier-Stokes-Fourier system), the
When additional external force is taken into account, the external force does affect the long time behavior of solutions. The upper bound of time decay rates were studied intensively, see for instance [1] and [2] on unbounded domain, [22], [23] on the convergence of the non-stationary flow to the corresponding steady flow when the initial date are small in
The main goal of current paper is to establish the sharp decay rate, on both upper and lower bounds, to the solutions of (1.1) using relatively simple energy method. We remark that similar results had been pursued by M. Schonbek [20], [21] for incompressible Navier-Stokes equations, and by Li, Matsumura-Zhang [13] for isentropic Navier-Stokes-Poisson system. Although they share the same spirit in obtaining the lower bound decay rates, the feature of the spectrum near zero exhibits quite different behaviors, leading to different analysis. For instance, we explored the elegant structure of the higher order nonlinear terms of Navier-Stokes, when choosing conservative variables: density and momentum. The conservative form of the sharp equations provided a natural derivative structure in these terms, leading to the possibility of a faster decay rate estimate. We will make a more detailed comparison later in this paper.
Define
{∂tn+divm=0,(t,x)∈R+×R3,∂tm+c2∇n−ˉμ△m−(ˉμ+ˉν)∇divm=F,(t,x)∈R+×R3,lim|x|→∞n=0,lim|x|→∞m=0,t∈R+,(n,m)|t=0=(ρ0−ˉρ,ρ0u0),x∈R3, | (1.2) |
where
F=−div{m⊗mn+ˉρ+ˉμ∇(nmn+ˉρ)}−∇{(ˉμ+ˉν)div(nmn+ˉρ)+(p(n+ˉρ)−p(ˉρ)−c2n)}. |
It is this structure of
Our aim is to obtain a clear picture of the large time behavior of
{∂t˜n+div˜m=0,(t,x)∈R+×R3,∂t˜m+c2∇˜n−ˉμ△˜m−(ˉμ+ˉν)∇div˜m=0,(t,x)∈R+×R3,lim|x|→∞˜n=0,lim|x|→∞˜m=0,t∈R+,(˜n,˜m)|t=0=(ρ0−ˉρ,ρ0u0),x∈R3, | (1.3) |
where
Notation. For
We now state our main result.
Theorem 1.1. Assume that
∫R3(n0,m0)dx≠0, | (1.4) |
then there is a unique global classical solution
C−1(1+t)−34−k2≤‖∇k˜n(t)‖L2(R3)≤C(1+t)−34−k2,k=0,1,2,3,C−1(1+t)−34−k2≤‖∇k˜m(t)‖L2(R3)≤C(1+t)−34−k2,k=0,1,2,3, |
and the initial value problem (1.2) has a unique solution
‖∇k(nh,mh)(t)‖L2(R3)≲δ20(1+t)−54−k2,k=0,1,2,‖∇3mh(t)‖L2(R3)≲δ20(1+t)−114,‖∇3nh(t)‖L2(R3)≲δ0(1+t)−74. |
As a consequence, there exists a positive constant
C−11(1+t)−34−k2≤‖∇kn(t)‖L2(R3)≤C1(1+t)−34−k2,k=0,1,2,C−11(1+t)−34−k2≤‖∇km(t)‖L2(R3)≤C1(1+t)−34−k2,k=0,1,2,3. |
Remark 1.1. We remark that this theorem is valid under the condition (1.4) which is important in the lower bound estimate to the linearized problem. When (1.4) fails, the decay rate of the linearized system (1.3) depends on the order of the degeneracy of moments. Assume
Remark 1.2. In [13], Li, Matsumura-Zhang proved the lower bound decay rate of the linearized isentropic Navier-Stokes-Poisson system, they only require
In what follows, we will set
{∂tn+ˉρdivu=−ndivu−u⋅∇n,∂tu+γˉρ∇n−ˉμ△u−(ˉμ+ˉν)∇divu=−u⋅∇u−ˉμf(n)△u−(ˉμ+ˉν)f(n)∇divu−g(n)∇n,lim|x|→∞n=0,lim|x|→∞u=0,(n,u)|t=0=(ρ0−ˉρ,u0), | (2.1) |
where
f(n):=nn+ˉρ,g(n):=p′(n+ˉρ)n+ˉρ−p′(ˉρ)ˉρ. | (2.2) |
We assume that there exist a time of existence
‖n(t)‖H3+‖u(t)‖H3≤δ, | (2.3) |
holds for any
ˉρ2≤n+ˉρ≤2ˉρ. |
Hence, we immediately have
|f(n)|,|g(n)|≤C|n|,|∇kf(n)|,|∇kg(n)|≤C∀k∈N+, | (2.4) |
where
Next, we begin with the energy estimates including
Theorem 2.1. Assume that
‖n0‖H3+‖u0‖H3≤δ0, |
then the problem (2.1) admits a unique global solution
‖n(t)‖2H3+‖u(t)‖2H3+∫t0(‖∇n(τ)‖2H2+‖∇u(τ)‖2H3)dτ≤C(‖n0‖2H3+‖u0‖2H3), |
where
The proof of this theorem is divided into several subsections.
For
12ddt∫R3(γ|n|2+|u|2)dx+∫R3(ˉμ|∇u|2+(ˉμ+ˉν)|divu|2)dx=∫R3γ(−ndivu−u⋅∇n)n−(u⋅∇u+ˉμf(n)△u+(ˉμ+ˉν)f(n)∇divu+g(n)∇n)⋅udx≲‖n‖L3‖∇u‖L2‖n‖L6+(‖u‖L3‖∇u‖L2+‖n‖L3‖∇n‖L2)‖u‖L6+(‖u‖L∞‖∇n‖L2+‖n‖L∞‖∇u‖L2)‖∇u‖L2≲(‖n‖L3+‖u‖L3+‖n‖L∞+‖u‖L∞)(‖∇n‖2L2+‖∇u‖2L2). | (2.5) |
Now for
12ddt∫R3(γ|∇n|2+|∇u|2)dx+∫R3(ˉμ|∇2u|2+(ˉμ+ˉν)|∇divu|2)dx≲(‖n‖L∞+‖u‖L∞+‖∇n‖L∞+‖∇u‖L∞)(‖∇n‖2L2+‖∇u‖2L2+‖∇2u‖2L2). | (2.6) |
For
12ddt∫R3(γ|∇2n|2+|∇2u|2)dx+∫R3(ˉμ|∇3u|2+(ˉμ+ˉν)|∇2divu|2)dx≲(‖n‖L∞+‖u‖L∞+‖∇n‖L∞+‖∇u‖L∞)(‖∇2n‖2L2+‖∇2u‖2L2+‖∇3u‖2L2). | (2.7) |
For
12ddt∫R3(γ|∇3n|2+|∇3u|2)dx+∫R3(ˉμ|∇4u|2+(ˉμ+ˉν)|∇3divu|2)dx≲(‖n‖L∞+‖u‖L∞+‖∇n‖L∞+‖∇u‖L∞)(‖∇3n‖2L2+‖∇3u‖2L2+‖∇4u‖2L2)+‖∇n‖L3‖∇4u‖L2‖∇2n‖L6+‖∇u‖L3‖∇4u‖L2‖∇2u‖L6+‖∇2n‖L3(‖∇3n‖L2+‖∇4u‖L2)‖∇2u‖L6. | (2.8) |
Summing up the above estimates, noting that
ddt∑0≤k≤3(γ‖∇kn‖2L2+‖∇ku‖2L2)+C1∑1≤k≤4‖∇ku‖2L2≤C2δ∑1≤k≤3‖∇kn‖2L2. | (2.9) |
For
ddt∫R3u⋅∇ndx+γˉρ∫R3|∇n|2dx≲‖∇u‖2L2+‖∇n‖L2‖∇2u‖L2+(‖n‖L∞+‖u‖L∞)(‖∇n‖2L2+‖∇u‖2L2), | (2.10) |
for
ddt∫R3∇u⋅∇2ndx+γˉρ∫R3|∇2n|2dx≲‖∇2u‖2L2+‖∇2n‖L2‖∇3u‖L2+(‖(n,u)‖L∞+‖(∇n,∇u)‖L∞)×(‖∇n‖2L2+‖∇2n‖2L2+‖∇2u‖2L2), | (2.11) |
and for
ddt∫R3∇2u⋅∇3ndx+γˉρ∫R3|∇3n|2dx≲‖∇3u‖2L2+‖∇3n‖L2‖∇4u‖L2+(‖(n,u)‖L∞+‖(∇n,∇u)‖L∞)×(‖∇2n‖2L2+‖∇2u‖2L2+‖∇3n‖2L2+‖∇3u‖2L2). | (2.12) |
Plugging the above estimates, using the smallness of
ddt∑0≤k≤2∫R3∇ku⋅∇k+1ndx+C3∑1≤k≤3‖∇kn‖2L2≤C4∑1≤k≤4‖∇ku‖2L2. | (2.13) |
Proof of Theorem 2.1. Multiplying (2.13) by
ddt{∑0≤k≤3(γ‖∇kn‖2L2+‖∇ku‖2L2)+2C2δC3∑0≤k≤2∫R3∇ku⋅∇k+1ndx}+C5{∑1≤k≤3‖∇kn‖2L2+∑1≤k≤4‖∇ku‖2L2}≤0. | (2.14) |
Next, we define
ddtE(t)+‖∇n(t)‖2H2+‖∇u(t)‖2H3≤0. | (2.15) |
Observe that since
C−16(‖n(t)‖2H3+‖u(t)‖2H3)≤E(t)≤C6(‖n(t)‖2H3+‖u(t)‖2H3). |
Then integrating (2.15) directly in time, we get
sup0≤t≤T(‖n(t)‖2H3+‖u(t)‖2H3)+C6∫T0(‖∇n(τ)‖2H2+‖∇u(τ)‖2H3)dτ≤C26(‖n0‖2H3+‖u0‖2H3). |
Using a standard continuity argument along with classical local wellposedness theory, this closes the a priori assumption (2.3) if we assume
In this section, we consider the initial value problem for the linearized Navier-Stokes system
{∂t˜n+div˜m=0,(t,x)∈R+×R3,∂t˜m+c2∇˜n−ˉμ△˜m−(ˉμ+ˉν)∇div˜m=0,(t,x)∈R+×R3,lim|x|→∞˜n=0,lim|x|→∞˜m=0,t∈R+,(˜n,˜m)|t=0=(ρ0−ˉρ,ρ0u0),x∈R3, | (3.1) |
where
In terms of the semigroup theory for evolutionary equations, the solution
˜Ut=B˜U,t≥0,˜U(0)=˜U0, |
which gives rise to
˜U(t)=S(t)˜U0=etB˜U0,t≥0, |
where
B=(0−div−c2∇ˉμ△angle+(ˉμ+ˉν)∇div). |
What left is to analyze the differential operator
∂tˆ˜U(t,ξ)=A(ξ)ˆ˜U(t,ξ),t≥0,ˆ˜U(0,ξ)=ˆ˜U0(ξ), |
where
A(ξ)=(0−iξt−c2iξ−ˉμ|ξ|2I3×3−(ˉμ+ˉν)ξ⊗ξ). |
The eigenvalues of the matrix
det(A(ξ)−λI)=−(λ+ˉμ|ξ|2)2(λ2+(2ˉμ+ˉν)|ξ|2λ+c2|ξ|2)=0, |
which implies
λ0=−ˉμ|ξ|2(double),λ1=λ1(|ξ|),λ2=λ2(|ξ|). |
The semigroup
etA=eλ0tP0+eλ1tP1+eλ2tP2, |
where the project operators
Pi=∏i≠jA(ξ)−λjIλi−λj. |
By a direct computation, we can verify the exact expression for the Fourier transform
ˆG(t,ξ)=etA=(λ1eλ2t−λ2eλ1tλ1−λ2−iξt(eλ1t−eλ2t)λ1−λ2−c2iξ(eλ1t−eλ2t)λ1−λ2e−λ0t(I−ξ⊗ξ|ξ|2)+ξ⊗ξ|ξ|2λ1eλ1t−λ2eλ2tλ1−λ2)=(ˆNˆM). |
Indeed, we can make the following decomposition for
ˆ˜n=ˆN⋅ˆ˜U0=(ˆN+ˆN)⋅ˆ˜U0,ˆ˜m=ˆM⋅ˆ˜U0=(ˆM+ˆM)⋅ˆ˜U0, |
where
ˆN=(λ1eλ2t−λ2eλ1tλ1−λ20),ˆN=(0−iξt(eλ1t−eλ2t)λ1−λ2),ˆM=(−c2iξ(eλ1t−eλ2t)λ1−λ20),ˆM=(0e−λ0t(I−ξ⊗ξ|ξ|2)+ξ⊗ξ|ξ|2λ1eλ1t−λ2eλ2tλ1−λ2). |
We further decompose the Fourier transform
Define
ˆN=ˆN1+ˆN2,ˆN=ˆN1+ˆN2,ˆM=ˆM1+ˆM2,ˆM=ˆM1+ˆM2, |
where
χ(ξ)={1,|ξ|≤R,0,|ξ|≥R+1. |
Then we have the following decomposition for
ˆ˜n=ˆN⋅ˆ˜U0=ˆN1⋅ˆ˜U0+ˆN2⋅ˆ˜U0=(ˆN1+ˆN1)⋅ˆ˜U0+(ˆN2+ˆN2)⋅ˆ˜U0,ˆ˜m=ˆM⋅ˆ˜U0=ˆM1⋅ˆU0+ˆM2⋅ˆ˜U0=(ˆM1+ˆM1)⋅ˆ˜U0+(ˆM2+ˆM2)⋅ˆ˜U0. | (3.2) |
To derive the long time decay rate of solution, we need to use accurate approximation to the Fourier transform
λ1=−2ˉμ+ˉν2|ξ|2+i2√4c2|ξ|2−(2ˉμ+ˉν)2|ξ|4=a+bi,λ2=−2ˉμ+ˉν2|ξ|2−i2√4c2|ξ|2−(2ˉμ+ˉν)2|ξ|4=a−bi, | (3.3) |
and we have
λ1eλ2t−λ2eλ1tλ1−λ2=e−12(2ˉμ+ˉν)|ξ|2t[cos(bt)+12(2ˉμ+ˉν)|ξ|2sin(bt)b]∼O(1)e−12(2ˉμ+ˉν)|ξ|2t,|ξ|≤η, |
λ1eλ1t−λ2eλ2tλ1−λ2=e−12(2ˉμ+ˉν)|ξ|2t[cos(bt)−12(2ˉμ+ˉν)|ξ|2sin(bt)b]∼O(1)e−12(2ˉμ+ˉν)|ξ|2t,|ξ|≤η, |
eλ1t−eλ2tλ1−λ2=e−12(2ˉμ+ˉν)|ξ|2tsin(bt)b∼O(1)1|ξ|e−12(2ˉμ+ˉν)|ξ|2t,|ξ|≤η, |
where
b=12√4c2|ξ|2−(2ˉμ+ˉν)2|ξ|4∼c|ξ|+O(|ξ|3),|ξ|≤η. |
For the high frequency
λ1=−2ˉμ+ˉν2|ξ|2−12√(2ˉμ+ˉν)2|ξ|4−4c2|ξ|2=a−b,λ2=−2ˉμ+ˉν2|ξ|2+12√(2ˉμ+ˉν)2|ξ|4−4c2|ξ|2=a+b, | (3.4) |
and we have
λ1eλ2t−λ2eλ1tλ1−λ2=12e(a+b)t[1+e−2bt]−a2be(a+b)t[1−e−2bt]∼O(1)e−R0t,|ξ|≥η, |
λ1eλ1t−λ2eλ2tλ1−λ2=a+b2be(a+b)t[1−e−2bt]+e(a−b)t∼O(1)e−R0t,|ξ|≥η, |
eλ1t−eλ2tλ1−λ2=12be(a+b)t[1−e−2bt]∼O(1)1|ξ|2e−R0t,|ξ|≥η, |
where
b=12√(2ˉμ+ˉν)2|ξ|4−4c2|ξ|2∼12(2ˉμ+ˉν)|ξ|2−2c22ˉμ+ˉν+O(|ξ|−2),|ξ|≥η. |
Here
In this section, we apply the spectral analysis to the semigroup for the linearized Navier-Stokes system. We will establish the
With the help of the formula for Green's function in Fourier space and the asymptotic analysis on its elements, we are able to establish the
Proposition 4.1. Let
‖∇k(˜n,˜m)(t)‖L2(R3)≤C(1+t)−34−k2(‖U0‖L1(R3)+‖∇kU0‖L2(R3)), |
where
Proof. A straightforward computation together with the formula of the Green's function
ˆ˜n(t,ξ)=λ1eλ2t−λ2eλ1tλ1−λ2ˆn0−iξ⋅ˆm0(eλ1t−eλ2t)λ1−λ2∼{O(1)e−12(2ˉμ+ˉν)|ξ|2t(|ˆn0|+|ˆm0|),|ξ|≤η,O(1)e−R0t(|ˆn0|+|ˆm0|),|ξ|≥η,ˆ˜m(t,ξ)=−c2iξ(eλ1t−eλ2t)λ1−λ2ˆn0+e−λ0tˆm0+(λ1eλ1t−λ2eλ2tλ1−λ2−e−λ0t)ξ(ξ⋅ˆm0)|ξ|2∼{O(1)e−ˉμ|ξ|2t(|ˆn0|+|ˆm0|),|ξ|≤η,O(1)e−R0t(|ˆn0|+|ˆm0|),|ξ|≥η, |
here and below,
‖(ˆ˜n,ˆ˜m)(t)‖2L2(R3)=∫|ξ|≤η|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ+∫|ξ|≥η|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ≲∫|ξ|≤ηe−2ˉμ|ξ|2t(|ˆn0|2+|ˆm0|2)dξ+∫|ξ|≥ηe−2R0t(|ˆn0|2+|ˆm0|2)dξ≲(1+t)−32‖(n0,m0)‖2L1(R3)∩L2(R3). |
And the
‖(^∇k˜n,^∇k˜m)(t)‖2L2(R3)=∫|ξ|≤η|ξ|2k|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ+∫|ξ|≥η|ξ|2k|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ≲∫|ξ|≤ηe−2ˉμ|ξ|2t|ξ|2k(|ˆn0|2+|ˆm0|2)dξ+∫|ξ|≥ηe−2R0t|ξ|2k(|ˆn0|2+|ˆm0|2)dξ≲(1+t)−32−k(‖(n0,m0)‖2L1(R3)+‖(∇kn0,∇km0)‖2L2(R3)). |
The proof of the Proposition 4.1 is completed.
It should be noted that the
Proposition 4.2. Let
C−1(1+t)−34−k2≤‖∇k˜n(t)‖L2(R3)≤C(1+t)−34−k2,C−1(1+t)−34−k2≤‖∇k˜m(t)‖L2(R3)≤C(1+t)−34−k2, |
where
Proof. We only show the case of
ˆ˜n(t,ξ)=λ1eλ2t−λ2eλ1tλ1−λ2ˆn0−iξ⋅ˆm0(eλ1t−eλ2t)λ1−λ2=e−12(2ˉμ+ˉν)|ξ|2t[cos(bt)ˆn0−iξ⋅ˆm0sin(bt)b]+e−12(2ˉμ+ˉν)|ξ|2t[12(2ˉμ+ˉν)|ξ|2sin(bt)bˆn0]=T1+T2,for|ξ|≤η, |
ˆ˜m(t,ξ)=−c2iξ(eλ1t−eλ2t)λ1−λ2ˆn0+e−λ0tˆm0+(λ1eλ1t−λ2eλ2tλ1−λ2−e−λ0t)ξ(ξ⋅ˆm0)|ξ|2=[e−12(2ˉμ+ˉν)|ξ|2t[cos(bt)ξ(ξ⋅ˆm0)|ξ|2−c2iξsin(bt)bˆn0]+e−ˉμ|ξ|2t[ˆm0−ξ(ξ⋅ˆm0)|ξ|2]]−e−12(2ˉμ+ˉν)|ξ|2t[12(2ˉμ+ˉν)|ξ|2sin(bt)bξ(ξ⋅ˆm0)|ξ|2]=S1+S2,for|ξ|≤η, |
here and below,
It is easy to check that
‖ˆ˜n(t,ξ)‖2L2=∫|ξ|≤η|ˆ˜n(t,ξ)|2dξ+∫|ξ|≥η|ˆ˜n(t,ξ)|2dξ≥∫|ξ|≤η|T1+T2|2dξ≥∫|ξ|≤η12|T1|2−|T2|2dξ. | (4.1) |
We then calculate that
\begin{equation} \begin{split} \int_{|\xi|\leq \eta}|T_2|^2 d\xi &\lesssim \|\widehat{n}_0\|_{L^\infty}^2\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}|\xi|^4\left(\frac{\sin(b t)}{b}\right)^2 d\xi \\ &\lesssim\|\widehat{n}_0\|_{L^\infty}^2\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}|\xi|^2 d\xi\lesssim(1+t)^{-\frac52}\|{n}_0\|_{L^1}^2. \end{split} \end{equation} | (4.2) |
Since
\begin{eqnarray*} \begin{split} |\widehat{n}_0(\xi)|^2\geq \frac1C\left|\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3}n_0(x) d x\right|^2 \geq \frac{M_n^2}C,\quad \text{for}\quad |\xi|\leq\eta. \end{split} \end{eqnarray*} |
For
\begin{eqnarray*} \begin{split} \frac{\left|\xi\cdot\widehat{m}_0(\xi)\right|^2}{|\xi|^2} \geq \frac{\left|\xi\cdot M_m\right|^2}{C|\xi|^2},\quad \text{for}\quad |\xi|\leq\eta. \end{split} \end{eqnarray*} |
When
\begin{equation} \label{optimal3} \begin{split} &\quad\int_{|\xi|\leq \eta}|T_1|^2 d\xi\nonumber\\ &\geq \frac{M_n^2}C\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\cos^2(b t)d\xi+\frac1C\int_{|\xi|\leq \eta}\frac{\left|\xi\cdot M_m\right|^2}{b^2}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\sin^2(b t)d\xi \nonumber \end{split} \end{equation} |
\begin{equation} \begin{split}&\geq \frac{\min\{{M_n^2},\frac{M_m^2}{3c^2}\}}C\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\big(\cos^2(b t)+\sin^2(b t)\big)d\xi\\ &\geq C_1\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}d\xi\\ &\geq C^{-1}(1+t)^{-\frac32}. \end{split} \end{equation} | (4.3) |
If
\begin{eqnarray*} |\widehat{m}_0(\xi)|^2 < \epsilon,\quad \text{for}\quad |\xi|\leq\eta. \end{eqnarray*} |
We thus use the help of spherical coordinates and the change of variables
\begin{equation} \begin{split} &\quad\int_{|\xi|\leq \eta}|T_1|^2 d\xi\\ &\geq \frac{M_n^2}C\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\cos^2(b t)d\xi-\frac{\epsilon}{Cc^2}\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\sin^2(b t)d\xi\\ &\geq \frac{M_n^2}Ct^{-\frac32}\int_0^{\eta\sqrt t}e^{-(2\bar\mu+\bar\nu)r^2}\cos^2(cr\sqrt t)r^2dr-\frac{\epsilon}{Cc^2}t^{-\frac32}\int_0^{\eta\sqrt t}e^{-(2\bar\mu+\bar\nu)r^2}\sin^2(cr\sqrt t)r^2dr\\ &\geq \frac{M_n^2}Ct^{-\frac32}\sum\limits_{k = 0}^{[\frac{c\eta t}\pi]-1}\int_{\frac{k\pi}{c\sqrt t}}^{\frac{k\pi+\frac\pi4}{c\sqrt t}}e^{-(2\bar\mu+\bar\nu)r^2}\cos^2(cr\sqrt t)r^2dr-\frac{\epsilon}{Cc^2}(1+t)^{-\frac32}\\ &\geq \frac{M_n^2}{2C}t^{-\frac32}\sum\limits_{k = 0}^{[\frac{c\eta t}\pi]-1}\int_{\frac{k\pi}{c\sqrt t}}^{\frac{k\pi+\frac\pi4}{c\sqrt t}}e^{-(2\bar\mu+\bar\nu)r^2}r^2dr-\frac{\epsilon}{Cc^2}(1+t)^{-\frac32}\\ &\geq C_1^{-1}(1+t)^{-\frac32}-C_2^{-1}\epsilon(1+t)^{-\frac32}\\ .&\geq C^{-1}(1+t)^{-\frac32} \end{split} \end{equation} | (4.4) |
In the case of
\begin{equation} \begin{split} &\quad\int_{|\xi|\leq \eta}|T_1|^2 d\xi\\ &\geq -\frac\epsilon C\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\cos^2(b t)d\xi+\frac{M_m^2}{3Cc^2}\int_{|\xi|\leq \eta}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\sin^2(b t)d\xi\\ &\geq C^{-1}(1+t)^{-\frac32}. \end{split} \end{equation} | (4.5) |
Combining the above estimates (4.1), (4.2), (4.3), (4.4) and (4.5), we obtain the lower bound of the time decay rate for
\begin{eqnarray*} \| {\widetilde n}(t,x)\|^2_{L^2} = \|\widehat {\widetilde n}(t,\xi)\|^2_{L^2}\geq C^{-1}(1+t)^{-\frac32}. \end{eqnarray*} |
The lower bound of the time decay rate for
\begin{equation} \begin{split} \|\widehat {\widetilde m}(t,\xi)\|^2_{L^2}\geq \int_{|\xi|\leq \eta}\frac12|S_1|^2-|S_2|^2 d\xi, \end{split} \end{equation} | (4.6) |
then we find that
\begin{equation} \begin{split} \int_{|\xi|\leq \eta}|S_2|^2 d\xi \lesssim(1+t)^{-\frac52}\|{m}_0\|_{L^1}^2. \end{split} \end{equation} | (4.7) |
We then calculate that
\begin{eqnarray*} \begin{split} &\quad\int_{|\xi|\leq \eta}|S_1|^2 d\xi\\ &\geq \bigg\{\frac{c^4M_n^2}C\int_{|\xi|\leq \eta}\frac{|\xi|^2}{b^2}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\sin^2(b t)d\xi\\ &\qquad+\frac{1}C\int_{|\xi|\leq \eta}\frac{\left|\xi\cdot M_m\right|^2}{|\xi|^2}e^{-(2\bar\mu+\bar\nu)|\xi|^2t}\cos^2(bt)d\xi\bigg\}\\ &\qquad +\bigg\{\int_{|\xi|\leq \eta}e^{-\frac12(4\bar\mu+\bar\nu)|\xi|^2t}\cos(bt)\frac{\xi(\xi\cdot{\widehat m}_0)}{|\xi|^2}\left({\widehat m}_0-\frac{\xi(\xi\cdot{\widehat m}_0)}{|\xi|^2}\right)d\xi\bigg\}\\ & = J_1+J_2. \end{split} \end{eqnarray*} |
A direct computation gives rise to
\begin{equation} J_1\geq C^{-1}(1+t)^{-\frac32},\qquad J_2 = 0. \end{equation} | (4.8) |
Combining the above estimates (4.6), (4.7) and (4.8), we obtain the lower bound of the time decay rate for
\begin{eqnarray*} \| {\widetilde m}(t,x)\|^2_{L^2} = \|\widehat {\widetilde m}(t,\xi)\|^2_{L^2}\geq C^{-1}(1+t)^{-\frac32}. \end{eqnarray*} |
Then the proof of Proposition 4.2 is completed.
In this subsection, we establish the following
Proposition 4.3. Let
\begin{eqnarray*} \|{\nabla}^{k} (\widetilde n,\widetilde m)(t)\|_{L^p({\mathop{\mathbb R\kern 0pt}\nolimits}^3)} \leq C(1+t)^{-\frac32(1-\frac1p)-\frac{k}2}\big(\|U_0\|_{L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^3)}+\|{\nabla}^k U_0\|_{L^p({\mathop{\mathbb R\kern 0pt}\nolimits}^3)}\big), \end{eqnarray*} |
where
To prove Proposition 4.3, the following two lemmas in [6] are helpful.
Lemma 4.1. Let
\begin{eqnarray*} \begin{split} |{\nabla}^\alpha _\xi \hat f(\xi)| \leq C'{ \left\{\begin{array}{l} |\xi|^{-|\alpha|+\sigma_1},\quad |\xi|\leq R, |\alpha| = n,\\ |\xi|^{-|\alpha|-\sigma_2},\quad |\xi|\geq R, |\alpha| = n-1,n,n+1, \end{array}\right.} \end{split} \end{eqnarray*} |
where
\begin{eqnarray*} f = m_1+m_2\delta, \end{eqnarray*} |
where
\begin{eqnarray*} m_2 = (2\pi)^{-\frac n 2}\lim\limits_{|\xi| \to \infty} \hat f(\xi), \end{eqnarray*} |
and
\begin{eqnarray*} \|f\ast g\|_{L^p} \leq C\| g\|_{L^p},\quad 1\leq p \leq \infty, \end{eqnarray*} |
where
Lemma 4.2. Let
\begin{eqnarray*} |{\nabla}_\xi ^\beta \hat f(\xi)|\leq C'|\xi|^{-|\beta|},\quad |\beta|\leq n+1. \end{eqnarray*} |
Then
\begin{eqnarray*} \|{\nabla}_x ^\alpha g(t,\cdot)\|_{L^p}\leq C(|\alpha|)t^{-\frac n 2(1-\frac 1 p)-\frac{|\alpha|}{2}}. \end{eqnarray*} |
In particular,
Now let us turn to the proof of Proposition 4.3.
Proof of Proposition 4.3. We first analyze above higher frequency terms denoted by
\begin{eqnarray*} \begin{split} \lambda_1 = -(2\bar\mu+\bar\nu) |\xi|^2+\frac{2c^2}{2\bar\mu+\bar\nu}+O(|\xi|^{-2}),\quad \lambda_2 = -\frac{2c^2}{2\bar\mu+\bar\nu}+O(|\xi|^{-2}),\quad |\xi|\geq \eta. \end{split} \end{eqnarray*} |
We shall prove that the higher frequency terms are
\begin{eqnarray*} \frac{\lambda_1e^{\lambda_2 t}-\lambda_2e^{\lambda_1 t}}{\lambda_1-\lambda_2} = e^{\lambda_2 t}+\frac{\lambda_2 e^{\lambda_2 t}}{\lambda_1-\lambda_2}-\frac{\lambda_2 e^{\lambda_1 t}}{\lambda_1-\lambda_2}. \end{eqnarray*} |
By a direct computation, it is easy to verify
\begin{eqnarray*} |{\nabla}_\xi^k \lambda_2|\lesssim|\xi|^{-2-k},\quad |\xi|\geq\eta, \end{eqnarray*} |
which gives rise to
\begin{eqnarray*} \begin{split} \bigg|{\nabla}_\xi^k \Big[(1-\chi(\cdot))e^{\lambda_2 t}\Big]\bigg|, \left|{\nabla}_\xi^k \Big[(1-\chi(\cdot))\frac{\lambda_2 e^{\lambda_2 t}}{\lambda_1-\lambda_2}\Big]\right|\lesssim{ \left\{\begin{array}{l} 0, \quad |\xi|\leq R,\\ e^{-c_1t}|\xi|^{-2-k},\quad |\xi|\geq R, \end{array}\right.} \end{split} \end{eqnarray*} |
here and below,
\begin{eqnarray*} (1-\chi(\cdot))\frac{\lambda_2 e^{\lambda_1 t}}{\lambda_1-\lambda_2} \sim e^{-\frac12(2\bar\mu+\bar\nu) |\xi|^2 t}\Big[(1-\chi(\cdot))\frac{e^{(-\lambda_2-\frac12(2\bar\mu+\bar\nu) |\xi|^2)t}}{\lambda_1-\lambda_2}\Big]. \end{eqnarray*} |
We can regard
\begin{equation} \|({\nabla}_x^k({\mathcal N}_2 \ast f),{\nabla}_x^k({\mathfrak N}_2 \ast f),{\nabla}_x^k( {\mathcal M}_2 \ast f),{\nabla}_x^k( {\mathfrak M}_2 \ast f))(t)\|_{L^p} \leq Ce^{-c_1t}\|{\nabla}_x^k f\|_{L^p}, \end{equation} | (4.9) |
for all integer
We also need to deal with the corresponding lower frequency terms denoted by
\begin{eqnarray*} \begin{split} &\frac{\lambda_1e^{\lambda_2 t}-\lambda_2e^{\lambda_1 t}}{\lambda_1-\lambda_2}, \frac{\lambda_1e^{\lambda_1 t}-\lambda_2e^{\lambda_2 t}}{\lambda_1-\lambda_2},\frac{|\xi|(e^{\lambda_1 t}-e^{\lambda_2 t})}{\lambda_1-\lambda_2}\sim O(1)e^{-\frac12 (2\bar\mu+\bar\nu)|\xi|^2t},\quad |\xi|\leq\eta, \end{split} \end{eqnarray*} |
which imply that for
\begin{eqnarray*} |\widehat{\mathcal N}_1|\sim O(1)e^{-c_2|\xi|^2t},\quad |\widehat{\mathfrak N}_1|\sim O(1)e^{-c_2|\xi|^2t},\\ |\widehat{\mathcal M}_1|\sim O(1)e^{-c_2|\xi|^2t},\quad |\widehat{\mathfrak M}_1|\sim O(1)e^{-c_2|\xi|^2t}, \end{eqnarray*} |
for some constants
\begin{equation} \begin{split} \|({\nabla}^k{\mathcal N}_1,{\nabla}^k{\mathfrak N}_1,{\nabla}^k {\mathcal M}_1,{\nabla}^k {\mathfrak M}_1)(t)\|_{L^p} \leq& C\left(\int_{|\xi|\leq \eta}\big||\xi|^k e^{-c_2|\xi|^2t}\big|^q d\xi\right)^{\frac 1q}\\ \leq& C(1+t)^{-\frac32(1-\frac1p)-\frac k2}. \end{split} \end{equation} | (4.10) |
Combining (4.9) and (4.10), we finally have for
\begin{eqnarray*} \begin{split} \|({\nabla}^k(N \ast f),{\nabla}^k(M \ast f))(t)\|_{L^p}& = \|({\nabla}^k((N_1+N_2) \ast f),{\nabla}^k((M_1+M_2) \ast f))(t)\|_{L^p}\\ &\leq C(1+t)^{-\frac32(1-\frac1p)-\frac k2}\|f\|_{L^1}+Ce^{-c_1t}\|{\nabla}^k f\|_{L^p}\\ &\leq C(1+t)^{-\frac32(1-\frac1p)-\frac k2}(\|f\|_{L^1}+\|{\nabla}^k f\|_{L^p}). \end{split} \end{eqnarray*} |
The proof of Proposition 4.3 is completed.
We are ready to prove Theorem 1.1 on the sharp time decay rate of the global solution to the initial value problem for the nonlinear Navier-Stokes system.
In what follows, we will set
\begin{equation} \left\{\begin{array}{l} \partial_t n_h + {\mathop{{\rm{div}}}\nolimits} m_h = 0,\qquad (t,x)\in{\mathop{\mathbb R\kern 0pt}\nolimits}^+\times{\mathop{\mathbb R\kern 0pt}\nolimits}^3,\\ \partial_t m_h + c^2 {\nabla} n_h- \bar\mu △ m_h- (\bar\mu+\bar\nu){\nabla}{\mathop{{\rm{div}}}\nolimits} m_h = F,\qquad (t,x)\in{\mathop{\mathbb R\kern 0pt}\nolimits}^+\times{\mathop{\mathbb R\kern 0pt}\nolimits}^3,\\ \lim\limits_{|x|\to\infty}n_h = 0, \quad\lim\limits_{|x|\to\infty} m_h = 0,\qquad t\in{\mathop{\mathbb R\kern 0pt}\nolimits}^+,\\ (n_h,m_h)\big|_{t = 0} = (0,0),\qquad x\in{\mathop{\mathbb R\kern 0pt}\nolimits}^3, \end{array}\right. \end{equation} | (5.1) |
where
\begin{eqnarray*} \begin{split} F = &-{\mathop{{\rm{div}}}\nolimits}\Big\{ \frac{(m_h+\widetilde m)\otimes (m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}+\bar\mu{\nabla}\big(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}\big)\Big\}\\ & - {\nabla}\Big\{(\bar\mu+\bar\nu){\mathop{{\rm{div}}}\nolimits}(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho})+\big(p(n_h+\widetilde n+\bar\rho)-p(\bar\rho)-c^2(n_h+\widetilde n)\big)\Big\}. \end{split} \end{eqnarray*} |
Denote
\begin{eqnarray*} \partial_t U_h = BU_h+H,\quad t\geq0,\qquad U_h(0) = 0, \end{eqnarray*} |
where the nonlinear term
\begin{eqnarray*} U_h(t) = S(t)\ast U_{h}(0)+\int_0^t S(t-\tau)\ast H(\widetilde U, U_h)(\tau)d\tau, \end{eqnarray*} |
which
\begin{equation} n_h = N\ast U_{h}(0)+\int_0^t \mathfrak N (t-\tau)\ast H(\tau)d\tau, \end{equation} | (5.2) |
\begin{equation} m_h = M\ast U_{h}(0)+\int_0^t \mathfrak M (t-\tau)\ast H(\tau)d\tau. \end{equation} | (5.3) |
Furthermore, in view of the above definition for
\begin{eqnarray*} |\widehat{\mathfrak N}(\xi)|\sim O(1)e^{-c_3|\xi|^2t}, \quad |\widehat{\mathfrak M}(\xi)|\sim O(1)e^{-c_3|\xi|^2t}, \quad|\xi|\leq\eta, \end{eqnarray*} |
\begin{eqnarray*} |\widehat{\mathfrak N}(\xi)|\sim O(1)\frac1{|\xi|}e^{-R_0t}, \quad |\widehat{\mathfrak M}(\xi)|\sim O(1)\frac1{|\xi|^2}e^{-R_0t}+O(1)e^{-c_4|\xi|^2t}, \quad |\xi|\geq\eta. \end{eqnarray*} |
Thus, applying a similar argument as in the proof of Proposition 4.1, we have
\begin{equation} \|({\nabla}^k {\mathfrak N}\ast H, {\nabla}^k {\mathfrak M}\ast H)(t)\|_{L^2} \leq C(1+t)^{-\frac32(\frac1q-\frac12)-\frac12-\frac k 2}\big(\|Q\|_{L^q}+\|{\nabla}^{k+1} Q\|_{L^2}\big),\quad q = 1,2, \end{equation} | (5.4) |
\begin{equation} \|({\nabla}^k {\mathfrak N}\ast H, {\nabla}^k {\mathfrak M}\ast H)(t)\|_{L^2} \leq C(1+t)^{-\frac32(\frac1q-\frac12)-\frac12-\frac k 2}\big(\|Q\|_{L^q}+\|{\nabla}^{k} Q\|_{L^2}\big),\quad q = 1,2, \end{equation} | (5.5) |
\begin{equation} \|{\nabla}^k {\mathfrak M}\ast H(t)\|_{L^2} \leq C(1+t)^{-\frac32(\frac1q-\frac12)-\frac12-\frac k 2}\big(\|Q\|_{L^q}+\|{\nabla}^{k-1} Q\|_{L^2}\big),\quad q = 1,2, \end{equation} | (5.6) |
for any non-negative integer
\begin{equation} \begin{split} Q = &\Big|\frac{(m_h+\widetilde m)\otimes (m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}+\bar\mu{\nabla}\big(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}\big)\Big|\\ &+\Big|(\bar\mu+\bar\nu){\mathop{{\rm{div}}}\nolimits}(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho})+\big(p(n_h+\widetilde n+\bar\rho)-p(\bar\rho)-c^2(n_h+\widetilde n)\big)\Big|. \end{split} \end{equation} | (5.7) |
For readers' convenience, we show how to estimate
\begin{eqnarray*} \begin{split} &\quad\|{\nabla}^k {\mathfrak M}\ast H(t)\|_{L^2}^2\\ &\lesssim\int_{|\xi|\leq \eta}e^{-2c_3|\xi|^2t}|\xi|^{2k}|\widehat H|^2d\xi +\int_{|\xi|\geq \eta}e^{-2R_0 t}|\xi|^{2k-4}|\widehat H|^2d\xi\\ &\quad+\int_{|\xi|\geq \eta}e^{-2c_4|\xi|^2 t}|\xi|^{2k}|\widehat H|^2d\xi\\ &\lesssim\int_{|\xi|\leq \eta}e^{-2c_3|\xi|^2t}|\xi|^{2k+2}|\widehat Q|^2d\xi +\int_{|\xi|\geq \eta}e^{-2R_0 t}|\xi|^{2k-2}|\widehat Q|^2d\xi\\ &\quad+\int_{|\xi|\geq \eta}e^{-2c_4|\xi|^2t}|\xi|^{2k+2}|\widehat Q|^2d\xi\\ &\lesssim(1+t)^{-3(\frac1q-\frac12)-1-k}\big(\|Q\|^2_{L^q({\mathop{\mathbb R\kern 0pt}\nolimits}^3)}+\|{\nabla}^{\tilde k} Q\|^2_{L^2({\mathop{\mathbb R\kern 0pt}\nolimits}^3)}\big),\quad q = 1,2,\quad k-1\leq\tilde k\in{\mathop{\mathbb N\kern 0pt}\nolimits}^+. \end{split} \end{eqnarray*} |
In this subsection, we establish the faster decay rate for
We begin with following Lemma.
Lemma 5.1. Let
\begin{eqnarray*} \begin{split} \int_0^{\frac t 2}(1+t-\tau)^{-r_1}(1+\tau)^{-r_2} d\tau = &\int_0^{\frac t 2}(1+\frac t 2+\tau)^{-r_1}(1+\frac t 2-\tau)^{-r_2} d\tau\\ \lesssim&{ \left\{\begin{array}{l} (1+t)^{-r_1}, \quad \mathit{\text{for}} \quad r_2 > 1,\\ (1+t)^{-(r_1-\epsilon)},\quad \mathit{\text{for}} \quad r_2 = 1,\\ (1+t)^{-(r_1+r_2-1)},\quad \mathit{\text{for}} \quad r_2 < 1, \end{array}\right.} \end{split} \end{eqnarray*} |
and
\begin{eqnarray*} \begin{split} \int_{\frac t 2}^t(1+t-\tau)^{-r_1}(1+\tau)^{-r_2} d\tau = &\int_0^{\frac t 2}(1+t-\tau)^{-r_2}(1+\tau)^{-r_1} d\tau\\ \lesssim&{ \left\{\begin{array}{l} (1+t)^{-r_2}, \quad \mathit{\text{for}} \quad r_1 > 1,\\ (1+t)^{-(r_2-\epsilon)},\quad \mathit{\text{for}} \quad r_1 = 1,\\ (1+t)^{-(r_1+r_2-1)},\quad \mathit{\text{for}} \quad r_1 < 1, \end{array}\right.} \end{split} \end{eqnarray*} |
where
Proposition 5.1. Under the assumptions of Theorem 1.1, the solution
\begin{eqnarray*} \begin{split} &\|({\nabla} ^k n_h,{\nabla}^{k} m_h)\|_{L^2}\leq C\delta_0^2(1+t)^{-\frac54-\frac {k} 2},\\ &\|{\nabla} ^3 m_h\|_{L^2}\leq C\delta_0^2(1+t)^{-\frac{11}4},\quad \|{\nabla} ^3 n_h\|_{L^2}\leq C\delta_0(1+t)^{-\frac74}, \end{split} \end{eqnarray*} |
where
From (5.7), we deduce
\begin{eqnarray*} Q(\widetilde U, U_h) = Q_1+Q_2+Q_3+Q_4, \end{eqnarray*} |
which implies for a smooth solution
\begin{eqnarray*} \begin{split} &Q_1 = Q_1(\widetilde U, U_h)\sim O(1)\left(n_h^2+m_h\otimes m_h+\widetilde n^2+\widetilde m\otimes\widetilde m \right),\\ &Q_2 = Q_2(\widetilde U, U_h)\sim O(1)\left(\widetilde n n_h+\widetilde m\otimes m_h\right),\\ &Q_3 = Q_3(\widetilde U, U_h)\sim O(1)\left({\nabla}(n_h\cdot m_h)+{\nabla}(\widetilde n\cdot\widetilde m)\right),\\ &Q_4 = Q_4(\widetilde U, U_h)\sim O(1)\left({\nabla}(\widetilde n\cdot m_h)+{\nabla}( n_h\cdot\widetilde m) \right). \end{split} \end{eqnarray*} |
Define
\begin{equation} \begin{split} \Lambda(t) = :&\sup\limits_{0\leq s \leq t}\bigg\{\sum\limits_{k = 0}^2(1+s)^{\frac54+\frac k 2}{\delta_0}^{-\frac34}\|({\nabla} ^k n_h,{\nabla}^{k}m_h)(s)\|_{L^2}\\ &\quad+(1+s)^{\frac74}\|({\nabla}^3 n_h, {\nabla}^3 m_h)(s)\|_{L^2}\bigg \}. \end{split} \end{equation} | (5.8) |
Proposition 5.2. Under the assumptions of Theorem 1.1, if for some
\begin{eqnarray*} \begin{split} \Lambda(t)\leq C\delta_0^{\frac34},\quad t\in[0,T], \end{split} \end{eqnarray*} |
where
The proof of this Proposition 5.2 consists of following three steps.
Starting with (5.4), (5.5), (5.6) and (5.8), we have after a complicate but straightforward computation that
\begin{equation} \begin{split} \|(n_h, m_h)\|_{L^2}&\lesssim\int_0^t \|(\mathfrak N (t-\tau)\ast H(\tau), \mathfrak M (t-\tau)\ast H(\tau))\|_{L^2}d\tau\\ &\lesssim\int_0^{t} (1+t-\tau)^{-\frac54}\big(\|Q(\tau)\|_{L^1}+\| Q(\tau)\|_{L^2}\big)d\tau\\ &\lesssim\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right)\int_0^{t} (1+t-\tau)^{-\frac54}(1+\tau)^{-\frac32}d\tau\\ &\lesssim(1+t)^{-\frac54}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{equation} | (5.9) |
It is easy to verify that
\begin{eqnarray*} \begin{split} \|Q(t)\|_{L^1}\lesssim&\|Q_1\|_{L^1}+\|Q_2\|_{L^1}+\|Q_3\|_{L^1}+\|Q_4\|_{L^1}\\ \lesssim &\|(\widetilde n,\widetilde m)\|_{L^2}^2+\|( n_h, m_h)\|_{L^2}^2+ \|( n_h, m_h)\|_{L^2}\big(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}\\ &+\|({\nabla} n_h,{\nabla} m_h)\|_{L^2}\big)+\|(\widetilde n,\widetilde m)\|_{L^2}\left(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2} +\| ({\nabla} n_h,{\nabla} m_h)\|_{L^2}\right)\\ \lesssim & (1+t)^{-\frac32}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{eqnarray*} |
Indeed, by virtue of Hölder's inequality and Gagliardo-Nirenberg's inequality, we obtain that
\begin{eqnarray*} \|u\|_{L^\infty}\lesssim \|{\nabla} u\|_{L^2}^{\frac12}\|{\nabla} ^2 u\|_{L^2}^{\frac12}, \end{eqnarray*} |
which implies that
\begin{eqnarray*} \begin{split} &\|Q(t)\|_{L^2}\\ \lesssim &\|(\widetilde n,\widetilde m)\|_{L^\infty}\big(\|(\widetilde n,\widetilde m)\|_{L^2}+\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}+\|(n_h,m_h)\|_{L^2}\\ &+\|({\nabla} n_h,{\nabla} m_h)\|_{L^2}\big)+\|( n_h,m_h)\|_{L^\infty}\left(\|( n_h,m_h)\|_{L^2}+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\right)\\ &+\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^\infty}\|(n_h,m_h)\|_{L^2}\\ \lesssim &(1+t)^{-\frac94}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{eqnarray*} |
Furthermore, exactly as in the estimate of the high order derivatives, we have
\begin{equation} \begin{split} &\|({\nabla} n_h, {\nabla} m_h)\|_{L^2}\\ \lesssim &\int_0^{\frac t 2} \|({\nabla}\mathfrak N , {\nabla}\mathfrak M)(t-\tau)\ast H(\tau)\|_{L^2}d\tau+\int_{\frac t 2}^t \|(\mathfrak N, \mathfrak M )(t-\tau)\ast {\nabla} H(\tau)\|_{L^2}d\tau\\ \lesssim &\int_0^{\frac t 2} (1+t-\tau)^{-\frac74}\big(\| Q(\tau)\|_{L^1}+\|{\nabla} Q(\tau)\|_{L^2}\big)d\tau +\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}\|{\nabla} Q(\tau)\|_{L^2}d\tau\\ \lesssim&\left(\delta_0^2+\delta_0^{\frac98}\Lambda^2(t)\right)\Bigg(\int_0^{\frac t 2} (1+t-\tau)^{-\frac74}(1+\tau)^{-\frac32}d\tau+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}(1+\tau)^{-\frac{11}4}d\tau\Big)\\ \lesssim&(1+t)^{-\frac74}\left(\delta_0^2+\delta_0^{\frac98}\Lambda^2(t)\right), \end{split} \end{equation} | (5.10) |
Similarly, it holds that
\begin{eqnarray*} \begin{split} &\|{\nabla} Q(t)\|_{L^2}\nonumber\\ \lesssim &\|(\widetilde n,\widetilde m)\|_{L^\infty}\big(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}+\|({\nabla}^2\widetilde n,{\nabla}^2\widetilde m)\|_{L^2}+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\nonumber \end{split} \end{eqnarray*} |
\begin{eqnarray*} \begin{split}&\quad+\|( {\nabla}^2 n_h,{\nabla}^2 m_h)\|_{L^2}\big)+\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^\infty}\big(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}+\|(n_h,m_h)\|_{L^2}\\ &\quad+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\big)+\|( n_h,m_h)\|_{L^\infty}\big(\|( {\nabla}^2\widetilde n,{\nabla}^2\widetilde m)\|_{L^2}+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\\ &\quad+\|({\nabla}^2 n_h,{\nabla}^2 m_h)\|_{L^2}\big)+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^\infty}\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\\ \lesssim & (1+t)^{-\frac{11}4}\left(\delta_0^2+\delta_0^{\frac98}\Lambda^2(t)\right). \end{split} \end{eqnarray*} |
Thus, we also get that
\begin{equation} \begin{split} &\|({\nabla}^2 n_h, {\nabla}^2 m_h)(t)\|_{L^2}\\ \lesssim& \int_0^{\frac t 2} \|({\nabla}^2 \mathfrak N, {\nabla}^2 \mathfrak M) (t-\tau)\ast H(\tau)\|_{L^2}d\tau\\ &\quad+\int_{\frac t 2}^t \|(\mathfrak N, \mathfrak M) (t-\tau)\ast {\nabla}^2 H(\tau)\|_{L^2}d\tau\\ \lesssim &\int_0^{\frac t 2}(1+t-\tau)^{-\frac94}\big(\|Q(\tau)\|_{L^1}+\|{\nabla}^2 Q(\tau)\|_{L^2}\big)d\tau\\ &\quad+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}\|{\nabla} ^2Q(\tau)\|_{L^2}d\tau\\ \lesssim& \left(\delta_0^2+\delta_0\Lambda(t)+\delta_0^{\frac34}\Lambda^2(t)\right)\bigg(\int_0^{\frac t 2} (1+t-\tau)^{-\frac94}(1+\tau)^{-\frac32}d\tau\\ &\quad+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}(1+\tau)^{-\frac{13}4}d\tau\bigg)\\ \lesssim&(1+t)^{-\frac94}\left(\delta_0^2+\delta_0\Lambda(t)+\delta_0^{\frac34}\Lambda^2(t)\right). \end{split} \end{equation} | (5.11) |
Finally, we have
\begin{eqnarray*} \begin{split} &\|{\nabla}^2 Q(t)\|_{L^2}\\ \lesssim &(\|(\widetilde n,\widetilde m)\|_{L^\infty}+\|(n_h,m_h)\|_{L^\infty})(\|({\nabla}^3\widetilde n,{\nabla}^3 \widetilde m)\|_{L^2}+\|({\nabla}^3n_h,{\nabla}^3 m_h)\|_{L^2})\\ &\quad+(\|({\nabla} \widetilde n,{\nabla}\widetilde m)\|_{L^\infty}+\|({\nabla} n_h,{\nabla} m_h)\|_{L^\infty}) (\|({\nabla}\widetilde n,{\nabla} \widetilde m)\|_{L^2}+\|({\nabla} n_h,{\nabla} m_h)\|_{L^2})\\ &\quad+(\|(\widetilde n,\widetilde m)\|_{L^\infty}+\|(n_h,m_h)\|_{L^\infty}+\|({\nabla} \widetilde n,{\nabla}\widetilde m)\|_{L^\infty}+\|({\nabla} n_h,{\nabla} m_h)\|_{L^\infty})\\ &\quad\times(\|({\nabla}^2\widetilde n,{\nabla}^2 \widetilde m)\|_{L^2}+\|({\nabla}^2 n_h,{\nabla}^2 m_h)\|_{L^2})\\ \lesssim&(1+t)^{-\frac{13}4}\left(\delta_0^2+\delta_0\Lambda(t)+\delta_0^{\frac34}\Lambda^2(t)\right). \end{split} \end{eqnarray*} |
In this subsection, we will close the a priori estimates and complete the proof of Proposition 5.2. For this purpose, we need to derive the time decay rate of higher order derivatives of
Lemma 5.2. Under the assumption of Theorem 1.1, one has
\begin{eqnarray*} \|{\nabla}^2 n(t)\|_{H^1}+\|{\nabla}^2 u(t)\|_{H^1}\lesssim (1+t)^{-\frac74}\left(\delta_0+\delta_0^{\frac34}\Lambda(t)\right). \end{eqnarray*} |
In particular, it holds that
\begin{eqnarray*} \|{\nabla}^3 (n_h, m_h)(t)\|_{L^2}\lesssim (1+t)^{-\frac74}\left(\delta_0+\delta_0^{\frac34}\Lambda(t)\right). \end{eqnarray*} |
Proof. First of all, in view of (2.12), recovering the dissipation estimate for
\begin{equation} \begin{split} &\frac{d}{dt}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^3 n dx +C_1\|{\nabla}^3 n\|_{L^2}^2 dx\\ \leq &C_2\left(\|{\nabla}^3 u\|_{L^2}^2+\|{\nabla}^4 u\|_{L^2}^2\right)+C(1+t)^{-\frac{3}2}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\\ &\quad\times\left(\|{\nabla}^2 n\|_{L^2}^2+\|{\nabla}^2 u\|_{L^2}^2+\|{\nabla}^3 u\|_{L^2}^2\right). \end{split} \end{equation} | (5.12) |
Summing up (2.7) and (2.8) in the energy estimate for
\begin{equation} \begin{split} &\frac{d}{dt}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} \left(\gamma |{\nabla}^2 n|^2+|{\nabla}^2 u|^2 + \gamma|{\nabla}^3 n|^2 +|{\nabla}^3 u|^2 \right)dx + C_3\left(\|{\nabla}^3 u|^2 _{L^2}+\|{\nabla}^4 u\|^2 _{L^2}\right) \\ \leq &C(1+t)^{-\frac{3}2}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\|{\nabla}^2 n\|_{L^2}^2+\|{\nabla}^2 u\|_{L^2}^2+\|{\nabla}^3 n\|_{L^2}^2\right). \end{split} \end{equation} | (5.13) |
Multiplying (5.12) by
\begin{eqnarray*} \begin{split} &\frac{d}{dt}\bigg\{\sum\limits_{2\leq k\leq3}\left(\gamma \|{\nabla}^k n\|^2_{L^2}+\|{\nabla}^k u\|^2_{L^2} \right)+\epsilon_1\frac{C_3}{C_2}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^{3} n dx \bigg\}\\ &\quad+ C_4\Big(\|{\nabla}^{3} n\|_{L^2}^2+\sum\limits_{3\leq k\leq4}\|{\nabla}^{k} u\|^2_{L^2}\Big)\\ \leq &C(1+t)^{-\frac{3}2}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\|{\nabla}^2 n\|_{L^2}^2+\|{\nabla}^2 u\|_{L^2}^2\right). \end{split} \end{eqnarray*} |
Next, we define
\mathcal E_1(t) = \bigg\{\sum\limits_{2\leq k\leq3}\left(\gamma \|{\nabla}^k n\|^2_{L^2}+\|{\nabla}^k u\|^2_{L^2} \right)+\epsilon_1\frac{C_3}{C_2}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^{3} n dx \bigg\}. |
Observe that since
\begin{eqnarray*} C_5^{-1}\left(\|{\nabla}^2 n(t)\|^2_{H^1}+\|{\nabla}^2 u(t)\|^2_{H^1}\right) \leq\mathcal E_1(t)\leq C_5\left(\|{\nabla}^2 n(t)\|^2_{H^1}+\|{\nabla}^2 u(t)\|^2_{H^1}\right). \end{eqnarray*} |
Then we arrive at
\begin{eqnarray*} \frac{d}{dt}\mathcal E_1(t)+C_4\Big(\|{\nabla}^{3} n(t)\|_{L^2}^2+\|{\nabla}^3 u(t)\|^2_{H^1}\Big) \leq C(1+t)^{-5}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{eqnarray*} |
Denote
\begin{eqnarray*} \begin{split} &\frac{C_4}{3}\|{\nabla}^{3} (n, u)(x)\|_{L^2}^2 \geq\frac{C_4}{3}\int_{S(t)^c} |\xi|^6|(\widehat{n}, \widehat{u})(\xi)|^2d\xi\\ \geq&(1+\gamma)(1+t)^{-1}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} |\xi|^4|(\widehat{n}, \widehat{u})(\xi)|^2d\xi-(1+\gamma)(1+t)^{-1}\int_{S(t)} |\xi|^4|(\widehat{n}, \widehat{u})(\xi)|^2d\xi. \end{split} \end{eqnarray*} |
Hence we have
\begin{eqnarray*} \begin{split} &\frac{d}{dt}\mathcal E_1(t)+(1+t)^{-1}\mathcal E_1(t)+\|{\nabla}^{3} n\|_{L^2}^2+\|{\nabla}^3 u\|^2_{H^1}\\ \lesssim&(1+t)^{-5}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right)+(1+t)^{-1}\int_{S(t)} |\xi|^4|(\widehat{n}, \widehat{u})(\xi)|^2d\xi\\ &\quad+(1+t)^{-1}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^{3} n dx. \end{split} \end{eqnarray*} |
Multiplying the above equation by
\begin{eqnarray*} \begin{split} &\frac{d}{dt}\Big\{(1+t)^5\mathcal E_1(t)\Big\}+(1+t)^5\Big(\|{\nabla}^{3} n\|_{L^2}^2+\|{\nabla}^3 u\|^2_{H^1}\Big) \lesssim(1+t)^{\frac12}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{eqnarray*} |
Integrating it with respect to time from
\begin{eqnarray*} \begin{split} &(1+t)^5\mathcal E_1(t)+\int_0^T(1+t)^5\Big(\|{\nabla}^{3} n\|_{L^2}^2+\|{\nabla}^3 u\|^2_{H^1}\Big)dt\\ \lesssim& \mathcal E_1(0)+(1+t)^{\frac32}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right), \end{split} \end{eqnarray*} |
which implies that
\begin{eqnarray*} \|{\nabla}^3 n\|^2_{L^2}+\|{\nabla}^3 u\|^2_{L^2}\lesssim\mathcal E_1(t)\lesssim (1+t)^{-5}\delta_0^2+(1+t)^{-\frac72}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{eqnarray*} |
Finally, we have
\begin{eqnarray*} \|{\nabla}^3 n_h\|_{L^2}+\|{\nabla}^3 m_h\|_{L^2}\lesssim (1+t)^{-\frac74}\left(\delta_0+\delta_0^{\frac34}\Lambda(t)\right). \end{eqnarray*} |
This completes the proof of this Lemma.
In this subsection, we first combine the above a priori estimates of (5.8), (5.9), (5.10), (5.11) and Lemma 5.2 together to give the proof of the Proposition 5.2. In deed, for any
\begin{equation} \Lambda(t)\leq C\left(\delta_0+\delta_0^{\frac14}\Lambda(t)+\Lambda^2(t)\right) \leq C\delta_0^{\frac34}. \end{equation} | (5.14) |
With the help of standard continuity argument, Proposition 5.2 and the smallness of
\begin{eqnarray*} \begin{split} &\|({\nabla}^k n_h, {\nabla}^k m_h)\|_{L^2}\lesssim \delta_0^2(1+t)^{-\frac54-\frac k2},\quad k = 0,1,\\ &\|{\nabla}^2 (n_h, m_h)\|_{L^2}\lesssim\delta_0^{\frac74}(1+t)^{-\frac94},\quad \|{\nabla}^3 (n_h, m_h)\|_{L^2}\lesssim\delta_0(1+t)^{-\frac74}. \end{split} \end{eqnarray*} |
Consequently, for any
\begin{equation} \Lambda(t)\leq C\delta_0. \end{equation} | (5.15) |
From (5.11) and (5.15), thus we also get that
\begin{eqnarray*} \|{\nabla}^2 (n_h, m_h)\|_{L^2}\lesssim\delta_0^2(1+t)^{-\frac94}. \end{eqnarray*} |
For
\begin{eqnarray*} \begin{split} &\|{\nabla}^3 m_h(t)\|_{L^2} \\\lesssim &\int_0^{\frac t 2}(1+t-\tau)^{-\frac{11}4}\big(\|Q(\tau)\|_{L^1}+\|{\nabla}^2 Q(\tau)\|_{L^2}\big)d\tau\\ &\quad+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}\|{\nabla} ^2Q(\tau)\|_{L^2}d\tau\\ \lesssim& \delta_0^2\bigg(\int_0^{\frac t 2} (1+t-\tau)^{-\frac{11}4}(1+\tau)^{-\frac32}d\tau+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}(1+\tau)^{-\frac{13}4}d\tau\bigg)\\ \lesssim&\delta_0^2(1+t)^{-\frac{11}4}. \end{split} \end{eqnarray*} |
Hence, we finish the proof of the Proposition 5.1. Theorem 1.1 follows.
Y. Chen is partially supported by the China Postdoctoral Science Foundation under grant 2019M663198, Guangdong Basic and Applied Basic Research Foundation under grant 2019A1515110733, NNSF of China under grants 11801586, 11971496 and China Scholarship Council. The research of R. Pan is partially supported by National Science Foundation under grants DMS-1516415 and DMS-1813603, and by National Natural Science Foundation of China under grant 11628103. L. Tong's research is partially supported by China Scholarship Council.
[1] |
Pandey SK, Patil SL, Ginoya D, Chaskar UM, Phadke SB (2019) Robust control of mismatched buck DC-DC converters by PWM-based sliding mode control schemes. Control Eng Pract 84: 183–193. https://doi.org/10.1016/j.conengprac.2018.11.010 doi: 10.1016/j.conengprac.2018.11.010
![]() |
[2] |
Zuo Wang S L, Jun Yang Q L (2018) Current sensorless finite-time control for buck converters with time-varying disturbances. Control Eng Pract 77: 127–137. https://doi.org/10.1016/j.conengprac.2018.05.014 doi: 10.1016/j.conengprac.2018.05.014
![]() |
[3] |
Wang JX, Rong JY, Li Y (2021) Reduced-order extended state observer based event-triggered sliding mode control for DC-DC buck converter system with parameter perturbation. Asian J Control 23: 1591–1601. http://doi.org/10.1002/asjc.2301 doi: 10.1002/asjc.2301
![]() |
[4] |
Wang B, Li S, Kan S, Li J (2023) Enhanced tracking of DC-DC buck converter systems using reduced-order extended state observer-based model predictive control. Int J Intell Syst 2: 143–152. https://doi.org/10.56578/jisc020303 doi: 10.56578/jisc020303
![]() |
[5] |
Oucheriah S (2024) Current-Sensorless Robust Sliding Mode Control for the DC-DC Buck Converter. Preprint at Research Square. https://doi.org/10.21203/rs.3.rs-4103291/v1 doi: 10.21203/rs.3.rs-4103291/v1
![]() |
[6] |
Cimini G, Ippoliti G, Orlando G, Longhi S, Miceli R (2017) A unified observer for robust sensorless control of DC-DC converters. Control Eng Pract 61: 21–27. https://doi.org/10.1016/j.conengprac.2017.01.012 doi: 10.1016/j.conengprac.2017.01.012
![]() |
[7] |
Pandey SK, Patil SL, Chaskar UM, Phadke SB (2019) State and Disturbance Observer-Based Integral Sliding Mode Controlled Boost DC-DC Converters. IEEE Trans Circuits Syst II Express Briefs 66: 1567–1571. https://doi.org/10.1109/TCSII.2018.2888570 doi: 10.1109/TCSII.2018.2888570
![]() |
[8] |
Malge SV, Patil SL, Chincholkar SH, Ghogare MG, Aher PK (2024) Inductor current estimation based sensorless control of boost type DC-DC converter. Control Eng Pract 153: 106119. https://doi.org/10.1016/j.conengprac.2024.106119 doi: 10.1016/j.conengprac.2024.106119
![]() |
[9] |
Malekzadeh M, Khosravi A, Tavan M (2019) A novel sensorless control scheme for DC-DC boost converter with global exponential stability. Eur Phys J Plus 134: 338. https://doi.org/10.1140/epjp/i2019-12664-4 doi: 10.1140/epjp/i2019-12664-4
![]() |
[10] |
Malekzadeh M, Khosravi A, Tavan M (2020) A novel adaptive output feedback control for DC-DC boost converter using immersion and invariance observer. Evol Syst 11: 707–715. https://doi.org/10.1007/s12530-019-09268-7 doi: 10.1007/s12530-019-09268-7
![]() |
[11] |
Zhang X, Martinez-Lopez M, He W, Shang Y, Jiang C, Moreno-Valenzuela J (2021) Sensorless Control for DC-DC Boost Converter via Generalized Parameter Estimation-Based Observer. Appl Sci 16: 7761. https://doi.org/10.3390/app11167761 doi: 10.3390/app11167761
![]() |
[12] |
Kim SK, Lee KB (2022) Current-Sensorless Energy-Shaping Output Voltage-Tracking Control for dc-dc Boost Converters With Damping Adaptation Mechanism. IEEE Trans Power Electron 37: 9266–9274. https://doi.org/10.1109/TPEL.2022.3159793 doi: 10.1109/TPEL.2022.3159793
![]() |
[13] |
Ayachit A, Kazimierczuk MK (2019) Averaged Small-Signal Model of PWM DC-DC Converters in CCM Including Switching Power Loss. IEEE Trans Circuits Syst II Express Briefs 66: 262–266. https://doi.org/10.1109/TCSII.2018.2848623 doi: 10.1109/TCSII.2018.2848623
![]() |
[14] |
Leon-Masich A, Valderrama-Blavi H, Bosque-Moncusi JM, Maixe-Altes J, Martinez-Salamero L (2015) Sliding-Mode-Control-Based Boost Converter for High-Voltage-Low-Power Applications. IEEE Trans Ind Electron 62: 229–237. https://doi.org/10.1109/TIE.2014.2327004 doi: 10.1109/TIE.2014.2327004
![]() |
[15] |
Martinez-Trevino BA, El Aroudi A, Valderrama-Blavi H, Cid-Pastor A, Vidal-Idiarte E, Martinez-Salamero L (2021) PWM Nonlinear Control With Load Power Estimation for Output Voltage Regulation of a Boost Converter With Constant Power Load. IEEE Trans Power Electron 36: 2143–2152. https://doi.org/10.1109/TPEL.2020.3008013 doi: 10.1109/TPEL.2020.3008013
![]() |
[16] |
Zambrano-Prada D, El Aroudi A, Vazquez-Seiszdedos L, Lopez-Santos O, Haroun R, Martinez-Salamero L (2023) Adaptive Sliding Mode Control for a Boost Converter with Constant Power Load. 2023 IEEE Conference on Power Electronics and Renewable Energy (CPERE), 1-6. https://doi.org/10.1109/CPERE56564.2023.10119573 doi: 10.1109/CPERE56564.2023.10119573
![]() |