In this paper, we consider maximum hands-off control problem governed by a nonlinear dynamical system, where the maximum hands-off control constraint is characterized by an L0 norm. For this problem, we first approximate the L0 norm constraint by a L1 norm constraint. Then, the control parameterization together with sequential adaptive switching time optimization technique is proposed to approximate the optimal control problem by a sequence of finite-dimensional optimization problems. Furthermore, a smoothing technique is exploited to approximate the non-smooth maximum operator and an error analysis is investigated for this approximation. The gradients of the cost functional with respect to the decision variables in the approximate problem are derived. On the basis of these results, we develop a gradient-based optimization algorithm to solve the resulting optimization problem. Finally, an example is solved to demonstrate the effectiveness of the proposed algorithm.
Citation: Sida Lin, Lixia Meng, Jinlong Yuan, Changzhi Wu, An Li, Chongyang Liu, Jun Xie. Sequential adaptive switching time optimization technique for maximum hands-off control problems[J]. Electronic Research Archive, 2024, 32(4): 2229-2250. doi: 10.3934/era.2024101
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In this paper, we consider maximum hands-off control problem governed by a nonlinear dynamical system, where the maximum hands-off control constraint is characterized by an L0 norm. For this problem, we first approximate the L0 norm constraint by a L1 norm constraint. Then, the control parameterization together with sequential adaptive switching time optimization technique is proposed to approximate the optimal control problem by a sequence of finite-dimensional optimization problems. Furthermore, a smoothing technique is exploited to approximate the non-smooth maximum operator and an error analysis is investigated for this approximation. The gradients of the cost functional with respect to the decision variables in the approximate problem are derived. On the basis of these results, we develop a gradient-based optimization algorithm to solve the resulting optimization problem. Finally, an example is solved to demonstrate the effectiveness of the proposed algorithm.
Of all the tourism segments, cultural tourism is, without a doubt, the one that has grown the most in the last decades. That is why this segment attracts not only the tourists' attention but that of the researchers too. Researchers have studied this segment from an economic, geographic and cultural point of view (Ferrari et al., 2018), considering the marked seasonality of the sector (Cuccia and Rizzo, 2011; Mondéjar-Jiménez et al., 2007). The tourist activity driven by the cultural and heritage elements (McKercher et al., 2005) provides an important source of income for their preservation. These market segments that have existed for years started to develop by the end of the last century (Ferrari et al., 2014). Doubtlessly, an important cultural and heritage offer means an improvement in the image of the destination and has a positive impact on the choice of the consumer (Castro et al., 2007, Pérez-Calderón et al., 2020). This has motivated the researchers to analyze this segment and to clearly define it in the scientific literature (Cetin and Bilgihan, 2016).
Many of these studies have focused on the cultural motivations of the tourists (Chhabra et al., 2003) to learn the profile and priorities when choosing a destination. However, they have particularly focused on assessing and knowing the main reasons that affect the choice of a destination and the opinion after its visit (Cordente-Rodríguez et al., 2011). This opinion can, doubtlessly, modify the initial idea and favor a return to the destination or its recommendation. Opinion has previously been studied in the literature (Yoon and Uysal, 2005). Nevertheless, there are few studies that manage to reflect, in a consistent manner, the previous motivations to the choice of destination. These motivations must be known in order to adapt the destination to the majority preferences of the visitors (McKercher, 2002; McKercher and du Cros, 2003) which, in many cases, are not managed properly (Garrod and Fyall, 2000).
Cultural tourism, although with different grades of motivation of the users (Tsiotsou and Vasaioti, 2006), encompasses several activities that, in many occasions, contribute to the main reason for visiting (McKercher and du Cros, 2003; Lavín et al., 2017). Segmenting the tourists according to new typologies can establish another research line to address in the future (Frochot, 2005; Poria et al., 2003).
The main objective of this study is assess a theoretical model to identify the main factors that determine the choosing of a tourist destination with an important cultural offer. The motivations of the tourists (parking, restaurants, transport...), together with the services the destination provides and its cultural offer, must affect its choice. The two factors under study shall alter these motivations as well.
The model proposed intends to measure the effects in different variables and know if the relations among them are statistically significant. For that purpose, the following hypotheses are stated:
H1: Tourists' motivations have a positive effect on the choice of a destination. Although this connection has been widely studied and previously contrasted (Chen and Rahman, 2018; Chhabra, 2009; Mondéjar-Jiménez and Vargas-Vargas, 2009; Mondéjar-Jiménez et al., 2009; Vergori and Arima, 2020; Xu et al., 2020), its intensity is the key factor to determine potential visitors.
H2: Services offered by the destination have a positive effect on the choice of said destination and the motivations of the potential tourists have been object of analysis and determine the image and subsequent choice of a tourist destination (García-Pozo et al., 2019; Hossein et al., 2019; Salazar, 2012; Scheyvens and Biddulph, 2018; Ying and Zhou, 2007).
√H2.1: Services offered by the destination have a positive effect on the choice of said destination.
√H2.2: Services offered by the destination have a positive effect on the motivations of the potential tourists.
H3: Cultural offer of the destination has a positive effect on the choice of said destination and on the motivations of its potential tourists (Hou et al., 2005; Lynch et al., 2011; Mondéjar-Jiménez et al., 2012; Richards, 2018; Secondi et al., 2011; Stylianou-Lambert, 2011). Cultural and heritage offer is possibly the main factor to determine a cultural destination.
√H3.1: Cultural offer of the destination has a positive effect on the choice of said destination.
√H3.2: Cultural offer of the destination has a positive effect on the motivations of the potential tourists.
These hypotheses shall be contrasted by means of a structural model using Partial Least Squares (PLS) due to its predictive character and the assumption of normality of some the variables (Assaker et al., 2013; Do Valle and Assaker, 2016; Ali et al., 2018; Hair et al., 2019).
PLS methodology based on variance structure is recommended for these initial models which, even though they are based on previous studies (Secondi et al., 2011), they also introduce some substantial changes. Models shall be adapted to the selected destination, in this case, the city of Malaga. This type of models shows a very solid performance (Henseler et al., 2009).
Partial Least Squares Path Modeling (PLS) technique analyzes the relation among the suggested latent variables which are measured through different items obtained in a survey of the city tourists. The relation among the latent variables can reflect direct and indirect effects. The sum of both effects is the total effect among latent variables. The number of observations is sufficient since they exceed the minimum needed for the model to function (Hair et al., 2019).
The city of Malaga has received more than 4.5 million visitors (tourists + hikers) with an economic impact of more than 3000 million euros, according to the city's tourist observatory. They have generated more than 2.5 million hotel overnight stays, with the United Kingdom, Germany and France being the main emitters of international tourists.
A study was conducted through interviews with visitors to the city of Malaga from June to December 2019. Given the great number of foreign visitors, the questionnaire was provided in Spanish and English. The questionnaire that was used was the one proposed in Mondéjar and Gómez (2009) which shows a breakdown by blocks of Motivations, Services and Cultural Offer of the visited destination.
Interviewed tourists were selected at random among the visitors of the main city monuments (the Cathedral of Malaga, the Alcazaba, Carmen Thyssen Museum and Picasso Museum) at the exit of their visit. A balance in places of origin, sex, age and nationality was pursued in order to obtain a sample as representative as possible. A total of 416 valid questionnaires have been used. Overall, 600 questionnaires were collected but many of them were not completely filled in. In some cases, because the tourist was in a hurry and could not wait to continue discovering the city; in other cases because it could be one of the first visited monuments and the tourist did not have a full idea of the services and cultural offer of the city.
A total of 18 indicators have been used to define 4 latent factors: Destination (2 indicators), Cultural Offer (3 indicators), Services (6 indicators) and Motivations (7 indicators). The Destination receives the direct effect of all the variables and the indirect effect of Services and Cultural Offer, which also has a direct effect on the Motivations of the tourists. Based on the block structure of the survey and the burden of each item, some have been suppressed from the original survey because they had a very low effect or did not meet the required minimum.
| Latent factor | Indicators | ||
| Tourist Destination | Destination valuation (V1) | Comparative valuation (V2) | |
| Cultural Offer | Kindness (R1) | Cultural offer (R2) | Heritage (R3) |
| City Services | Cleaning (S1)Safety (S4) | Preservation (S2)Green areas (S5) | Sign posts (S3)Access (S6) |
| Tourist Motivations | Visiting (M4)Nature (M7)Festivals (M10) | Languages (M5)Beach (M8) | Sport (M6)Relax (M9) |
DownLoad:
CSV
The SmartPLS 3 software has been used to estimate the model. The established relations among the latent variables can be observed in Figure 1.
Table 2 displays the results with the burdens for each indicator, all the recommended weight being sufficient in every latent factor except the factor Motivations. This is largely due to the heterogeneity of the sample and the different starting points of the tourists since, in some cases, the visitors already knew the destination.
| Culture | Motivations | Destination | Services | |
| M4 | 0.165 | 0.537 | 0.179 | 0.163 |
| M5 | 0.291 | 0.579 | 0.131 | 0.270 |
| M6 | 0.185 | 0.641 | 0.276 | 0.233 |
| M7 | 0.233 | 0.521 | 0.123 | 0.134 |
| M8 | 0.122 | 0.334 | 0.131 | 0.081 |
| M9 | 0.100 | 0.313 | 0.045 | 0.056 |
| M10 | 0.125 | 0.488 | 0.263 | 0.132 |
| R1 | 0.764 | 0.342 | 0.315 | 0.446 |
| R2 | 0.794 | 0.279 | 0.292 | 0.484 |
| R3 | 0.765 | 0.198 | 0.323 | 0.544 |
| S1 | 0.432 | 0.182 | 0.327 | 0.690 |
| S2 | 0.502 | 0.215 | 0.309 | 0.642 |
| S3 | 0.315 | 0.187 | 0.135 | 0.525 |
| S4 | 0.458 | 0.238 | 0.310 | 0.783 |
| S5 | 0.384 | 0.292 | 0.335 | 0.667 |
| S6 | 0.390 | 0.177 | 0.175 | 0.631 |
| V1 | 0.346 | 0.362 | 0.907 | 0.421 |
| V2 | 0.360 | 0.242 | 0.846 | 0.311 |
DownLoad:
CSV
The correlation matrix of latent variables is shown in the following table, the highest correlation being between the Cultural Offer and the Services provided by the destination.
| Culture | Motivations | Destination | Services | |
| Culture | 1 | |||
| Motivations | 0.3590 | 1 | ||
| Destination | 0.4004 | 0.3509 | 1 | |
| Services | 0.6303 | 0.3325 | 0.4235 | 1 |
DownLoad:
CSV
Table 4 shows the different reliability measures of the model. The obtained R-Square values are high in both cases and the analysis of internal consistency is acceptable (Cronbach's Alpha) except for the latent factor Motivations, as might be expected in the light of the results in Table 2. It is the same with the composite reliability indices. It is in the Motivations variable where no value is higher than 0.8.
| AVE | CompositeReliability | R Square | Cronbach's Alpha | Communality | Redundancy | |
| Culture | 0.599 | 0.817 | 0.668 | 0.599 | ||
| Motivations | 0.250 | 0.689 | 0.147 | 0.520 | 0.250 | 0.029 |
| Destination | 0.768 | 0.869 | 0.245 | 0.702 | 0.768 | 0.082 |
| Services | 0.436 | 0.821 | 0.743 | 0.436 |
DownLoad:
CSV
Table 5 shows the different reliability measures of the model. The obtained R-Square values are high in both cases and the analysis of internal consistency is acceptable (Cronbach's Alpha) except for the latent factor Motivations, as might be expected in the light of the results in Table 2. It is the same with the composite reliability indices. It is in the Motivations variable where no value is higher than 0.8. shows the hypothesis test for the direct effects, with a significance level of 95% in all cases. Consequently, all null hypotheses and the positive effects that the Motivations, Culture and Services variables have when choosing a destination can be confirmed. Even though Services has the greater intensity, the results align with previous studies. Indeed, the Services of the destination condition the choice together with the diverse Motivations of the tourist; to a lesser extent the Cultural Offer, since in many cases the tourist does not enjoy it fully, depending on the duration of the stay.
| Total Effects | T Statistics | |
Culture  Motivations |
0.247 | 2.838* |
| Culture Destination |
0.221 | 2.706* |
| Motivations Destination |
0.207 | 3.000* |
| Services Motivations |
0.176 | 1.947 |
| Services Destination |
0.247 | 3.423* |
| Note: *Significant values at the 5% significance level. | ||
DownLoad:
CSV
The first of the conclusions is the validity and reliability of the proposed model which, given its predictive character, can be used by public and private agents for policy-making and destination improvement. Thus, some of the lowest-rated indicators, specially sign posts and safety, can be improved.
The second conclusion is related to the improvement of motivations of tourists when choosing a destination. The lowest rated are Beach and Relax. It is true that Malaga is not the main exponent of the Spanish Costa del Sol (one of the best valued sun-and-sand destinations globally) but the city of Malaga has magnificent unknown beaches that must be promoted. Furthermore, it is a destination where one can get high levels of relaxation, which is another motivation to choose it. It is certain that the exponential growth in tourists, caused mainly by the arrival of cruises, has turn Malaga into a mass destination. Nevertheless, this massification is concentrated in the historic center adjacent to the port and cruise terminals, leaving the rest of the city as a quiet place where one can enjoy.
The third conclusion is related to the significant increase in cultural offer in the city of Malaga in the last 20 years. Out of the many sporting, cultural and leisure events, we must highlight the approach of Malaga, together with Madrid, as a national referent of "City of Museums". In fact, it has one of the most important offers (at a national and European level) with a total of 40 museums, the most relevant being the following: Picasso Museum, Museum of the City of Malaga, Carmen Thyssen Malaga Museum, Malaga Pompidou Centre and Russian Museum Malaga.
In this paper, we showed the relative impact that the different tourist resources have on the election of a given destination. Undoubtedly, the strategic planning of the city must consider these results so as to improve them in future years. All the hypotheses and subhypotheses have been confirmed (influence of the Services in the Motivations, only at a 90%). Therefore, we must emphasize direct effects on the latent variable Destination. However, we should highlight that the effects have a similar result, with values around 0.25.
We must take into account that, among the future research lines to address, one would be the segmentation of the respondents by means of sex or place of residence (national/foreign). We believe that, in both cases, motivations shall be different, and the perception of services and cultural offer could suffer fluctuations.
All authors declare no conflicts of interest in this paper.
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Figures(2) / Tables(1)
Sida Lin, Lixia Meng, Jinlong Yuan, Changzhi Wu, An Li, Chongyang Liu, Jun Xie. Sequential adaptive switching time optimization technique for maximum hands-off control problems[J]. Electronic Research Archive, 2024, 32(4): 2229-2250. doi: 10.3934/era.2024101
| Latent factor | Indicators | ||
| Tourist Destination | Destination valuation (V1) | Comparative valuation (V2) | |
| Cultural Offer | Kindness (R1) | Cultural offer (R2) | Heritage (R3) |
| City Services | Cleaning (S1)Safety (S4) | Preservation (S2)Green areas (S5) | Sign posts (S3)Access (S6) |
| Tourist Motivations | Visiting (M4)Nature (M7)Festivals (M10) | Languages (M5)Beach (M8) | Sport (M6)Relax (M9) |
DownLoad:
CSV
| Culture | Motivations | Destination | Services | |
| M4 | 0.165 | 0.537 | 0.179 | 0.163 |
| M5 | 0.291 | 0.579 | 0.131 | 0.270 |
| M6 | 0.185 | 0.641 | 0.276 | 0.233 |
| M7 | 0.233 | 0.521 | 0.123 | 0.134 |
| M8 | 0.122 | 0.334 | 0.131 | 0.081 |
| M9 | 0.100 | 0.313 | 0.045 | 0.056 |
| M10 | 0.125 | 0.488 | 0.263 | 0.132 |
| R1 | 0.764 | 0.342 | 0.315 | 0.446 |
| R2 | 0.794 | 0.279 | 0.292 | 0.484 |
| R3 | 0.765 | 0.198 | 0.323 | 0.544 |
| S1 | 0.432 | 0.182 | 0.327 | 0.690 |
| S2 | 0.502 | 0.215 | 0.309 | 0.642 |
| S3 | 0.315 | 0.187 | 0.135 | 0.525 |
| S4 | 0.458 | 0.238 | 0.310 | 0.783 |
| S5 | 0.384 | 0.292 | 0.335 | 0.667 |
| S6 | 0.390 | 0.177 | 0.175 | 0.631 |
| V1 | 0.346 | 0.362 | 0.907 | 0.421 |
| V2 | 0.360 | 0.242 | 0.846 | 0.311 |
DownLoad:
CSV
| Culture | Motivations | Destination | Services | |
| Culture | 1 | |||
| Motivations | 0.3590 | 1 | ||
| Destination | 0.4004 | 0.3509 | 1 | |
| Services | 0.6303 | 0.3325 | 0.4235 | 1 |
DownLoad:
CSV
| AVE | CompositeReliability | R Square | Cronbach's Alpha | Communality | Redundancy | |
| Culture | 0.599 | 0.817 | 0.668 | 0.599 | ||
| Motivations | 0.250 | 0.689 | 0.147 | 0.520 | 0.250 | 0.029 |
| Destination | 0.768 | 0.869 | 0.245 | 0.702 | 0.768 | 0.082 |
| Services | 0.436 | 0.821 | 0.743 | 0.436 |
DownLoad:
CSV
| Total Effects | T Statistics | |
| Culture Motivations |
0.247 | 2.838* |
| Culture Destination |
0.221 | 2.706* |
| Motivations Destination |
0.207 | 3.000* |
| Services Motivations |
0.176 | 1.947 |
| Services Destination |
0.247 | 3.423* |
| Note: *Significant values at the 5% significance level. | ||
DownLoad:
CSV
| Latent factor | Indicators | ||
| Tourist Destination | Destination valuation (V1) | Comparative valuation (V2) | |
| Cultural Offer | Kindness (R1) | Cultural offer (R2) | Heritage (R3) |
| City Services | Cleaning (S1)Safety (S4) | Preservation (S2)Green areas (S5) | Sign posts (S3)Access (S6) |
| Tourist Motivations | Visiting (M4)Nature (M7)Festivals (M10) | Languages (M5)Beach (M8) | Sport (M6)Relax (M9) |
| Culture | Motivations | Destination | Services | |
| M4 | 0.165 | 0.537 | 0.179 | 0.163 |
| M5 | 0.291 | 0.579 | 0.131 | 0.270 |
| M6 | 0.185 | 0.641 | 0.276 | 0.233 |
| M7 | 0.233 | 0.521 | 0.123 | 0.134 |
| M8 | 0.122 | 0.334 | 0.131 | 0.081 |
| M9 | 0.100 | 0.313 | 0.045 | 0.056 |
| M10 | 0.125 | 0.488 | 0.263 | 0.132 |
| R1 | 0.764 | 0.342 | 0.315 | 0.446 |
| R2 | 0.794 | 0.279 | 0.292 | 0.484 |
| R3 | 0.765 | 0.198 | 0.323 | 0.544 |
| S1 | 0.432 | 0.182 | 0.327 | 0.690 |
| S2 | 0.502 | 0.215 | 0.309 | 0.642 |
| S3 | 0.315 | 0.187 | 0.135 | 0.525 |
| S4 | 0.458 | 0.238 | 0.310 | 0.783 |
| S5 | 0.384 | 0.292 | 0.335 | 0.667 |
| S6 | 0.390 | 0.177 | 0.175 | 0.631 |
| V1 | 0.346 | 0.362 | 0.907 | 0.421 |
| V2 | 0.360 | 0.242 | 0.846 | 0.311 |
| Culture | Motivations | Destination | Services | |
| Culture | 1 | |||
| Motivations | 0.3590 | 1 | ||
| Destination | 0.4004 | 0.3509 | 1 | |
| Services | 0.6303 | 0.3325 | 0.4235 | 1 |
| AVE | CompositeReliability | R Square | Cronbach's Alpha | Communality | Redundancy | |
| Culture | 0.599 | 0.817 | 0.668 | 0.599 | ||
| Motivations | 0.250 | 0.689 | 0.147 | 0.520 | 0.250 | 0.029 |
| Destination | 0.768 | 0.869 | 0.245 | 0.702 | 0.768 | 0.082 |
| Services | 0.436 | 0.821 | 0.743 | 0.436 |
| Total Effects | T Statistics | |
| Culture Motivations |
0.247 | 2.838* |
| Culture Destination |
0.221 | 2.706* |
| Motivations Destination |
0.207 | 3.000* |
| Services Motivations |
0.176 | 1.947 |
| Services Destination |
0.247 | 3.423* |
| Note: *Significant values at the 5% significance level. | ||
