Stability of multi-population traffic flows

1.   Introduction

The main goal of this paper is to study one class of optimal control problems (OCPs) for a viscous Boussinesq system arising in the study of the dynamics of cardiovascular networks. We consider the boundary control problem for a $1D$ system of coupled PDEs with the Robin-type boundary conditions, describing the dynamics of pressure and flow in the arterial segment. We discuss in this part of paper the existence of optimal solutions and provide a substantial analysis of the first-order optimality conditions. Namely, we deal with the following minimization problem:

subject to the constraints

and

Here, $\beta_g$, $\beta_h$, and $\eta^\ast$ are positive constants, and $G_{ad}$ and $H_{ad}$ are the sets of admissible boundary controls. These sets and the rest of notations will be specified in the next section.

Optimal control problem (1)-(5) comes from the fluid dynamic models of blood flows in arterial systems. It is well known that the cardiovascular system consists of a pump that propels a viscous liquid (the blood) through a network of flexible tubes. The heart is one key component in the complex control mechanism of maintaining pressure in the vascular system. The aorta is the main artery originating from the left ventricle and then bifurcates to other arteries, and it is identified by several segments (ascending, thoracic, abdominal). The functionality of the aorta, considered as a single segment, is worth exploring from a modeling perspective, in particular in relationship to the presence of the aortic valve.

In the first part of our investigation (see [5]) we make use of the standard viscous hyperbolic system (see [2,21]) which models cross-section area $S(x,t)$ and average velocity $u(x,t)$ in the spatial domain:

where $(t,x)\in Q = (0,L)\times(0,T)$, $f = f(x,t)$ is a friction force, usually taken to be $f = -22\mu \pi u/S$, $\mu$ is the fluid viscosity, $P(x,t)$ is the hydrodynamic pressure, $L$ is the length of an arterial segment, and $T = T_{pulse} = 60/(Hart Rate)$ is the duration of an entire heartbeat. Here we include the inertial effects of the wall motion, described by the wall displacement $\eta = \eta (x,t)$:

where $r(x,t)$ is the radius, $r_{0} = r(x,0),$ $S_{0} = S(x,0)$.

The fluid structure interaction is modeled using inertial forces, which gives the pressure law

Here, $P_{ext}$ is the external pressure, $\beta = \frac{E}{1-\sigma ^{2}}h$, $\sigma$ is the Poisson ratio (usually $\sigma^2 = \frac{1}{2}$), $E$ is Young modulus, $h$ is the wall thickness, $m = \frac{\rho_{\omega }h}{2\sqrt{\pi S_{0}}}$, $\rho _{\omega }$ is the density of the wall.

This leads to the following Boussinesq system:

where $\rho$ is the blood density. Considering the relation $\eta_{t} = -\frac 12 r_0 u_{x}$ and rearranging terms in $u$ we get the system in the form (2)-(3). It remains to furnish the system by corresponding initial and boundary conditions which we propose to take in the form (3)-(4).

As for the OCP that is related with the arterial system, we are interested in finding the optimal heart rate (HR) which leads to the minimization of the following cost functional

The systolic period is taken to be consistently one quarter of $T_{pulse}$, and $P_{ref} = 100$ mmHg.

It is easy to note that relations (8)-(9) lead to the following representation for the cost functional (10)

Since $\eta_t\approx -\frac{1}{2}r_0 u_x$ (see [3]) and we suppose that $\nu\eta_{xx}$ should be small enough, it easily follows from (11) that the given cost functional (10) can be reduced to the tracking type (1).

The research in the field of the cardiovascular system is very active (see, for instance the literature describing the dynamics of the vascular network coupled with a heart model, [2,9,10,12,15,16,17,18,19,20,21]). However, there seems to be no studies that focus on both aspects at the same time: a detailed description of the four chambers of the heart and on the spatial dynamics in the aorta. Some numerical aspects of optimizing the dynamics of the pressure and flow in the aorta as well as the heart rate variability, taking into account the elasticity of the aorta together with an aortic valve model at the inflow and a peripheral resistance model at the outflow, based on the discontinuous Galerkin method and a two-step time integration scheme of Adam-Bashfort, were recently treated in [3] for the Boussinesq system like (2). More broadly, theory and applications of optimization and control in spatial networks, basing on the different types of conservation laws have been extensively developed in literature, have been successfully applied to telecommunications, transportation or supply networks ([6,7]).

From mathematical point of view, the characteristic feature of the Boussinesq system (2) is the fact that it involves a pseudo-parabolic operator with unbounded coefficient in its principle part. In the first part of this paper it was shown that for any pair of boundary controls $g\in G_{ad}$ and $h\in H_{ad}$, and for given $f\in L^\infty(0,T;L^2(\Omega))$, $\mu\in L^\infty(0,T;L^2(\Omega))$, $\sigma_0\in L^\infty(0,T)$, $\sigma_1\in L^\infty(0,T)$, $u_0\in V_\delta$, $\eta_0\in H_0^1(\Omega)$, $r_0\in H^1(\Omega)$, and $\delta\in L^1(\Omega)$, the set of feasible solutions to optimal control problem (1)-(5) is non-empty and the corresponding weak solution $(\eta(t),u(t))$ of the viscous Boussinesq system (2)-(4) possesses the extra regularity properties $\eta_{xx}, u_{xt}\in L^2(0,T;L^2(\Omega))$, which play a crucial role in the proof of solvability of OCP (1)-(5). In this paper we deal with the existence of optimal solutions and derive the corresponding optimality conditions for the problem (1)-(5). It should be mentioned, that application of Lagrange principle requires even higher smoothness of solutions to the initial Boussinesq system (2)-(4). In order to avoid such limitations, we deal with a simplified version of the initial optimal control problem (2)-(4) (see (39), argumentation above and [3,5] for physical description of the considered model). Also, in the second part of the paper, in order to provide the thorough substantiation of the first-order optimality conditions to the considered OCP, we make the special assumption for $\delta$ to be an element of the class $H^1(\Omega)$. Since the coefficient $\delta$ depends on such indicators as wall thickness, density of the wall and blood density, i.e. indicators varying slowly and smoothly, such assumption seems justified.

2.   Preliminaries

Let $T>0$ and $L>0$ be given values. We set $\Omega = (0,L)$, $Q = (0,T)\times\Omega$, and $\Sigma = (0,T)\times \partial\Omega$. Let $\delta\in H^1(\Omega)$ be a given function such that $\delta(x)\ge \delta_0>0$ for a.e. $x\in \Omega$. We use the standard notion $L^2(\Omega,\delta\,dx)$ for the set of measurable functions $u$ on $\Omega$ such that

We set $H = L^2(\Omega)$, $V_0 = H^1_0(\Omega)$, $V = H^1(\Omega)$, and identify the Hilbert space $H$ with its dual $H^\ast$. On $H$ we use the common natural inner product $(\cdot,\cdot)_H$, and endow the Hilbert spaces $V_0$ and $V$ with the inner products

and

respectively.

We also make use of the weighted Sobolev space $V_\delta$ as the set of functions $u\in V$ for which the norm

is finite. Note that due to the following estimate, $V_\delta$ is complete with respect to the norm $\|\cdot\|_{V,\delta}$:

Recall that $V_0$, $V$, and, hence, $V_\delta$ are continuously embedded into $C(\overline{\Omega})$, see [1,14] for instance. Since $\delta,\delta^{-1}\in L^1(\Omega)$, it follows that $V_\delta$ is a uniformly convex separable Banach space [14]. Moreover, in view of the estimate (12), the embedding $V_\delta\hookrightarrow H$ is continuous and dense. Hence, $H = H^\ast$ is densely and continuously embedded in $V^\ast_\delta$, and, therefore, $V_\delta\hookrightarrow H\hookrightarrow V^\ast_\delta$ is a Hilbert triplet (see [11] for the details).

Let us recall some well-known inequalities, that will be useful in the sequel (see [5]).

● $\|u\|_{L^\infty(\Omega)}\le \sqrt{2\max\left\{L,L^{-1}\right\}} \|u\|_{V}$, $\forall\, u\in V$ and $\|u\|_{L^\infty(\Omega)}\le 2\sqrt{L}\,\|u\|_{V_0}$, $\forall\, u\in V_0$.

● (Friedrich's Inequality) For any $u\in V_0$, we have

By $L^2(0,T;V_0)$ we denote the space of measurable abstract functions (equivalence classes) $u:[0,T]\rightarrow V$ such that

By analogy we can define the spaces $L^2(0,T;V_\delta)$, $L^\infty(0,T;H)$, $L^\infty(0,T;V_\delta)$, and $C([0,T];H)$ (for the details, we refer to [8]). In what follows, when $t$ is fixed, the expression $u(t)$ stands for the function $u(t,\cdot)$ considered as a function in $\Omega$ with values into a suitable functional space. When we adopt this convention, we write $u(t)$ instead of $u(t,x)$ and $\dot{u}$ instead of $u_t$ for the weak derivative of $u$ in the sense of distribution

where $\left<\cdot,\cdot\right>_{V^\ast;V}$ denotes the pairing between $V^\ast$ and $V$.

We also make use of the following Hilbert spaces

supplied with their common inner product, see [8,p. 473], for instance.

Remark 1. The following result is fundamental (see [8]): Let $(V,H,V^\ast)$ be a Hilbert triplet, $V\hookrightarrow H\hookrightarrow V^\ast$, with $V$ separable, and let $u\in L^2(0,T;V)$ and $\dot{u}\in L^2(0,T;V^\ast)$. Then

(ⅰ) $u\in C([0,T];H)$ and $\exists\,C_E>0$ such that

(ⅱ) if $v\in L^2(0,T;V)$ and $\dot{v}\in L^2(0,T;V^\ast)$, then the following integration by parts formula holds:

for all $s,t\in[0,T]$.

The similar assertions are valid for the Hilbert triplet $V_\delta\hookrightarrow H\hookrightarrow V^\ast_\delta$.

3.   On solvability of optimal control problem (1)-(5)

Let $\nu>0$ be a viscosity parameter, and let

be given distributions. In particular, $f$ stands for a fixed forcing term, $u_\Omega$ and $\eta_Q$ are given desired states for the wall displacement and average velocity, respectively, $\alpha_\Omega$ and $\alpha_Q$ are non-negative weights (without loss of generality we suppose that $\alpha_Q$ is a nonnegative constant function on $[0,T]\times[0,L]$), $u_0$ and $\eta_0$ are given initial states, and $\delta$ is a singular (possibly locally unbounded) weight function such that $\delta(x)\ge \delta_0>0$ for a.e. $x\in \Omega$.

We assume that the sets of admissible boundary controls $G_{ad}$ and $H_{ad}$ are given as follows

where $g_0,h_0,g_1,h_1\in L^\infty(0,T)$ with $g_0(t)\le g_1(t)$ and $h_0(t)\le h_1(t)$ almost everywhere in $(0,T)$.

The optimal control problem we consider in this paper is to minimize the discrepancy between the given distributions $(u_\Omega,\eta_Q)\in L^2(\Omega)\times L^2(Q)$ and the pair of distributions $\left(u(T),\eta(t)+\eta_{tt}(t)\right)$ (see, for instance, [5] for the physical interpretation), where $(\eta(t),u(t))$ is the solution of a viscous Boussinesq system, by an appropriate choice of boundary controls $g\in G_{ad}$ and $h\in H_{ad}$. Namely, we deal with the minimization problem (1)-(5).

Definition 3.1. We say that, for given boundary controls $g\in G_{ad}$ and $h\in H_{ad}$, a couple of functions $(\eta(t),u(t))$ is a weak solution to the initial-boundary value problem (2)-(4) if

and the following relations

hold true for all $\varphi\in V_0$ and $\psi\in V_\delta$ and a.e. $t\in[0,T]$.

Remark 2. Let us mention that if we multiply the left- and right-hand sides of equations (23)-(24) by function $\chi\in L^2(0,T)$ and integrate the result over the interval $(0,T)$, all integrals are finite. Moreover, closely following the arguments of Korpusov and Sveshnikov (see [13]), it can be shown that the weak solution to (2)-(4) in the sense of Definition 3.1 is equivalent to the following one: $(\eta(t),u(t))$ is a weak solution to the initial-boundary value problem (2)-(4) if the conditions (19)-(22) hold true and

where

Lemma 3.2 ([5]). Assume that the conditions (15)-(17) hold true. Let $g\in G_{ad}$ and $h\in H_{ad}$ be an arbitrary pair of admissible boundary controls. Then there exists a unique solution $(\eta(\cdot),u(\cdot))$ of the system (2)-(4) in the sense of Definition 3.1 such that

and there exists a constant $D_\ast>0$ depending only on initial data (15), (17) and control constrains $h_1, g_1$, satisfying the estimates

We also define the feasible set to the problem (1)-(5), (18) as follows:

We say that a tuple $\left(g^0,h^0,\eta^0,u^0\right)\in \Xi$ is an optimal solution to the problem (1)-(5), (18) if

In [5] it was shown that $\Xi\ne \emptyset$ and $\Xi_\lambda = \left\{(g,h,\eta,u)\in\Xi:J(g,h,\eta,u)\le\lambda\right\}$ is a bounded set in $L^2(0,T)\times L^2(0,T)\times\left(W_0(0,T)+\eta^\ast\right)\times W_\delta(0,T)$ for every $\lambda>0$.

While proving these hypotheses, the authors in [5] obtained a series of useful estimates for the weak solutions to initial-boundary value problem (2)-(4).

Lemma 3.3. [5,Lemmas 6.3 and 6.5 along with Remark 6.5] Let $g\in G_{ad}$ and $h\in H_{ad}$ be an arbitrary pair of admissible boundary controls. Let $(\eta(\cdot),u(\cdot)) = (w(\cdot)+\eta^\ast,u(\cdot))$ be the corresponding weak solution to the system (2)-(4) in the sense of Definition 3.1. Under assumptions (15)-(17), there exist positive constants $C_1$, $C_2$, $C_3$ depending on the initial data only such that for a.a. $t\in [0,T]$

In the context of solvability of OCP (18)-(5), the regularity of the solutions of the corresponding initial-boundary value problem (2)-(4) plays a crucial role.

Theorem 3.4 ([5]). The set of feasible solutions $\Xi$ to the problem (1)-(5), (18) is nonempty provided the initial data satisfy the conditions (15)-(17).

Now we proceed with the result concerning existence of optimal solutions to OCP (1)-(5), (18).

Theorem 3.5. For each

the optimal control problem (1)-(5), (18) admits at least one solution $(g^0,h^0,\eta^0,u^0)$.

Proof. We apply for the proof the direct method of the calculus of variations. Let us take $\lambda \in\mathbb{R}_+$ large enough, such that

Since the cost functional (1) is bounded below on $\Xi$, this implies the existence of a minimizing sequence $\{(g_n,h_n,\eta_n,u_n)\}_{n\ge \mathbb{N}}\subset \Xi_\lambda$, where $\eta_n = w_n+\eta^\ast$. In [5], the authors have proved that this sequence is bounded in $L^2(0,T)\times L^2(0,T)\times \left(W_0(0,T)+\eta^\ast\right)\times W_\delta(0,T)$. Moreover, using (30)-(31), we get

Therefore, within a subsequence, still denoted by the same index, we can suppose that

where $v = \dot{u}^0$ in the sense of distributions $\mathcal{D}^\prime(0,T;V_\delta)$. Also, by Lemma 3.3 (see relation (33)), we get

whence, passing to a subsequence, if necessary, we obtain

due to the continuity of embedding $V_\delta\hookrightarrow V$ and the compactness of embedding $V\hookrightarrow H$. In view of this, lower semicontinuity of norms in $L^2(0,T)$, $L^2(\Omega)$ with respect to the weak convergence and the fact that

we have $J(g^0,h^0,\eta^0,u^0)\le\inf_{n\in\mathbb{N}}J(g_n,h_n,\eta_n,u_n)$.

4.   Auxiliary results

This section aims to prove a range of auxiliary results that will be used in the sequel. Throughout this section the tuple $(g^0,h^0,\eta^0,u^0)$, where $\eta^0 = w^0+\eta^\ast$ denotes an optimal solution to initial OCP problem (1)-(5).

The following proposition aims to prove rather technical result, however it is useful for substantiation of the first-order optimality conditions to the initial OCP (1)-(5).

Proposition 1. Let $\delta\in H^1(\Omega)$. Then, for the initial data (15)-(17), the following inclusions take place

Proof. To begin with, let us prove that

Obviously, in order to show that

it would be enough to apply the similar arguments. Since $\eta^0\in W(0,T;V)\hookrightarrow C(Q)$, it is enough to show that there exists $\widetilde{C}$ such that

It should be noticed that as far as

then $u^0_{xx}\in (H^1(\Omega))^\ast = V^\ast$.

Also the fact that $\eta^0\in H^2(\Omega)$ gives $\eta^0_{xx}\in L^2(\Omega)$ and $\eta^0_x\in H^1(\Omega)\hookrightarrow C(\overline{\Omega})$ for a.a. $t\in[0;T]$. Therefore, we have

It is clear that if only $\eta^0\in \left(W_0(0,T)+\eta^\ast\right)\cap L^2(0,T;H^2(\Omega)\cap V)$, then we have $\eta^0\in C(0,T;V)$, $\eta^0\in C(\overline{\Omega})$, and $\eta^0_x\in L^2(0,T;V)$. Moreover, from $(\delta u_x^0)_x = \delta_xu^0_x+\delta u^0_{xx}$ we can deduce that

and

Let us show that the integrals $\int_0^T\|\delta_x u^0_x\|^2_{V^\ast}\,dt$ and $\int_0^T \|(\delta u_x^0)_x\|^2_{V^\ast}\,dt$ are finite. We take into account the continuous embedding $V\hookrightarrow C(\overline{\Omega})$. Then $\exists\, c(E)$ such that $\|v\|_{C(\overline{\Omega})}\le c(E)\|v\|_{V}$, for all $v\in V$. As for the first integral, we have

Now, to estimate the second integral, we make use of the equation (2)$_2$ and the well known inequality $(a+b+c)^2\le 3(a^2+b^2+c^2)$.

It is worth to mention here that, in fact, $(\delta (u_0)_x)_x\in (H^1(\Omega))^\ast$ because the element $\delta (u_0)_x$ belongs to $L^2(\Omega)$. Indeed,

It remains to note that the property $\int_0^T\left(\int_\Omega \left(\eta^0-\eta_Q\right)\,dx\right)^2 \,dt<\infty$ can be rewritten as follows $\int_\Omega \left(\eta^0-\eta_Q\right)\,dx\in L^2(0,T)$.

Let us consider two operators $\gamma_1$ and $\gamma_2$ that define the restriction of the functions from $V = H^1(\Omega)$ to the boundary $\partial\Omega = \left\{x = L,x = 0\right\}$, respectively (i.e. $\gamma_1[u(t,\cdot)] = u(t,L)$ and $\gamma_2[u(t,\cdot)] = u(t,0)$). Also we put into consideration two operators

defined on the set of vector functions ${\bf{p}} = (p,q)^t\in L^2(0,T;V_0)\times L^2(0,T;V_\delta)$ by the rule

Here, we use the fact that $V_\delta^\ast = V_0^\ast\oplus H^{-1/2}(\partial\Omega)$, which in one-dimensional case obviously turns to $V^\ast = V_0^\ast\oplus \mathbb{R}\oplus \mathbb{R}$ and, hence, $L^2(0,T;V_\delta^\ast) = L^2(0,T;V_0)\oplus L^2(0,T)\oplus L^2(0,T)$. Then the following result holds true.

Lemma 4.1. The operator $A(t):V_0\times V_\delta\to \left[V_0^\ast\right]^2\times\mathbb{R}\times\mathbb{R}$, defined by (36), satisfies the following conditions:

$A(t)$ is radially continuous, i.e. for any fixed ${\bf{v}}_1,\,{\bf{v}}_2\in V_0\times V_\delta: = \widetilde{V}$ and almost each $t\in(0,T)$ the real-valued function $s\rightarrow \langle A(t)({\bf{v}}_1+s {\bf{v}}_2),{\bf{v}}_2\rangle_{\widetilde{V}^\ast;\widetilde{V}}$ is continuous in $[0,1]$;

for some constant $C$ and some function $g\in L^2(0,T)$

it is strictly monotone uniformly with respect to $t\in [0,T]$ in the following sense: there exists a constant $m>0$, independent of $t$, such that

Moreover, the operator $B:L^2(0,T;V_0)\times L^2(0,t;V_\delta)\to \left[L^2(0,T;V_0^\ast)\right]^2\times L^2(0,T)\times L^2(0,T)$ possesses the Lipschitz property, i.e. there exists a constant $L>0$ such that

Proof. Since the radial continuity of operator $A$ is obvious, we begin with the proof of the second property. Let ${\bf{v}} = (v,w),\,{\bf{z}} = (z,y)\in \widetilde{V}$ be arbitrary elements. Then

As for the monotonicity property, for every ${\bf{p}}_1,{\bf{p}}_2\in V_0\times V_\delta$, we have

It remains to show the Lipschitz continuity of operator $B(t)$. With that in mind we consider three vector-valued functions ${\bf{v}} = (v_1,v_2)^t$, ${\bf{w}} = (w_1,w_2)^t$ and ${\bf{z}} = (z_1,z_2)^t$. Then

Taking into account the continuous embedding $V_\delta,V_0\hookrightarrow C(\overline{\Omega})$ and the corresponding inequality

we finally have

where $L = \max\{C_1;C_2\}$ and

This concludes the proof.

Lemma 4.2. Operator

which is defined by (36), is radially continuous, strictly monotone and there exists an inverse Lipschitz-continuous operator

such that

where $A^{-1}(t): \left[V_0^\ast\right]^2\times\mathbb{R}\times\mathbb{R}\to V_0\times V_\delta$ is an inverse operator to

Proof. It is easy to see that the action of operator $A(t)$ on element ${\bf{p}} = (p,q)^t$ can be also given by the rule:

where

It is easy to see, that $A_1(t)$ is the identity operator. Therefore, $A_1^{-1}(t)\equiv A_1(t)$. As for the operator $A_2(t)$, it is strongly monotone for all $t\in [0,T]$ because

Moreover, $A_2(t)$ satisfies all preconditions of [11,Lemma 2.2] that establishes the existence of a Lipschitz continuous inverse operator

such that

where $A_2^{-1}(t): \left[V_0^\ast\right]\times\mathbb{R}\times\mathbb{R}\to V_\delta$ is an inverse operator to $A_2(t): V_\delta\to V_0^\ast\times\mathbb{R}\times\mathbb{R}$. The proof is complete.

Before proceeding further, we make use of the following result concerning the solvability of Cauchy problems for pseudoparabolic equations (for the proof we refer to [11,Theorem 2.4]).

Theorem 4.3. For operators

defined in (36), (37), and for any

the Cauchy problem

admits a unique solution.

5.   First-order optimality conditions

In this section we focus on the derivation of the first-order optimality conditions for optimization problem (1)-(5). The Lagrange functional

associated to problem (1)-(5) (see also Remark 2) is defined by

Let us shift the correspondent derivatives from $w$ and $u$ to Lagrange multipliers $p$ and $q$, taking into account the initial and boundary conditions (3)-(4):

For each fixed $(p,q)\in \Big(W_0(0,T)\cap L^2(0,T;H^2(\Omega)\cap V_0)\Big)\times W^{1,\infty}(0,T;V_\delta)$ the Lagrangian is continuously Frechet-differentiable with respect to

Notice that, for a fixed $t$, we have $u\in V\subset C(\overline{\Omega})$ and $w\in V_0\subset C(\overline{\Omega})$, hence, the inner products $(w_x(t)u(t)+w(t)u_x(t),p(t))_{H}$ and $(u(t) u_x(t),q(t))_H$ are correctly defined almost everywhere in $[0,T]$.

Further we make use of the following relation $\eta_t = -\frac{1}{2}r_0 u_x$ that was introduced in [3]. Substituting this one to (2), we have $\nu\eta_{xx} = (\eta u)_x = \eta_x u+u_x\eta$.

Also, to simplify the deduction and in order to avoid the demanding of the increased smoothness on solutions of the initial Boussinesq system (2)-(5), we use (see [4] and [5]) elastic model for the hydrodynamic pressure

instead of the inertial one

Indeed, if we suppose the wall thickness $h$ to be small enough, the last term in the inertial model (38) appears negligible.

As a result, the cost functional $J(g,h,w,u)$, where $\eta = w+\eta^\ast$, takes the form

In order to formulate the conjugate system for an optimal solution $(g^0,h^0,\eta^0,u^0)$, where $\eta^0 = w^0+\eta^\ast$, we have to find the Fréchet differentials $\mathcal{L}_w z$ and $\mathcal{L}_u v$, where

With that in mind we emphasize the following point. Since the elements

are some admissible solutions to OCP (39), (2)-(5), it follows that the increments $z$ and $v$ satisfy the homogeneous initial and boundary conditions, i.e.

Taking into account the definition of the Fréchet derivative of nonlinear mappings, we get

where $R_0(w,z)$ stands for the remainder, which takes the form

and

It is obviously follows from (42) that

Hence, after some transformations, we have

Treating similarly to the other derivative, we obtain

where the remainder $\widetilde{\mathcal{R}}_0(u,v)$ takes the form

and

We are now in a position to identify the Fréchet derivatives $\mathcal{L}_w$ and $\mathcal{L}_v$ of the Lagrangian. Following in a similar manner, we have

and

As for the Fréchet derivatives $\mathcal{L}_g$ and $\mathcal{L}_h$, direct calculations leads us to the following representation:

where

Taking into account the calculations given above, we arrive at the following representation of the first-order optimality conditions for OCP (2)-(5), (39).

Theorem 5.1. Let $(g^0,h^0,\eta^0,u^0)$, where $\eta^0 = w^0+\eta^\ast$, be an optimal solution to the optimal control problem (1)-(5). Then there exists a unique pair

such that the following system

holds true for all

and a.e. $t\in[0,T]$.

Proof. Since the derived optimality conditions (46)-(55) are the direct consequence of the Lagrange principle, we focus on the solvability of the variational problems (48)-(49) for the adjoint variables $p$ and $q$. To do so, we represent the system (48)-(49) as the corresponding equalities in the sense of distributions, namely,

In the operator presentation, the system (56)-(62) takes the form (see [11]):

where the operators

are defined in (36)-(37), and

As a result, the existence of a unique pair $(p(t),q(t))$ satisfying the system (48)-(51) is a mere consequence of Theorem 5.1. Moreover, since the Cauchy problem has a solution for any

the Lagrange multiplier $\lambda$ in the definition of the Lagrange functional

can be taken equal to $1$.