{"title":"Stability analysis for abstract theomoelastic systems with Cattaneo's law and inertial terms","authors":"","doi":"10.3934/mcrf.2022053","DOIUrl":"https://doi.org/10.3934/mcrf.2022053","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86251459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive feedback control laws for isolation, contact regulation, and vaccination for infectious diseases, using a strict Lyapunov function. We use an SIQR epidemic model describing transmission, isolation via quarantine, and vaccination for diseases to which immunity is long-lasting. Assuming that mass vaccination is not available to completely eliminate the disease in a time horizon of interest, we provide feedback control laws that drive the disease to an endemic equilibrium. We prove the input-to-state stability (or ISS) robustness property on the entire state space, when the immigration perturbation is viewed as the uncertainty. We use an ISS Lyapunov function to derive the feedback control laws. A key ingredient in our analysis is that all compartment variables are present not only in the Lyapunov function, but also in a negative definite upper bound on its time derivative. We illustrate the efficacy of our method through simulations, and we discuss the usefulness of parameters in the controls. Since the control laws are feedback, their values are updated based on data acquired in real time. We also discuss the degradation caused by the delayed data acquisition occurring in practical implementations, and we derive bounds on the delays under which the ISS property is ensured when delays are present.
{"title":"Feedback control of isolation and contact for SIQR epidemic model via strict Lyapunov function","authors":"H. Ito, Michael A. Malisoff, F. Mazenc","doi":"10.3934/mcrf.2022043","DOIUrl":"https://doi.org/10.3934/mcrf.2022043","url":null,"abstract":"We derive feedback control laws for isolation, contact regulation, and vaccination for infectious diseases, using a strict Lyapunov function. We use an SIQR epidemic model describing transmission, isolation via quarantine, and vaccination for diseases to which immunity is long-lasting. Assuming that mass vaccination is not available to completely eliminate the disease in a time horizon of interest, we provide feedback control laws that drive the disease to an endemic equilibrium. We prove the input-to-state stability (or ISS) robustness property on the entire state space, when the immigration perturbation is viewed as the uncertainty. We use an ISS Lyapunov function to derive the feedback control laws. A key ingredient in our analysis is that all compartment variables are present not only in the Lyapunov function, but also in a negative definite upper bound on its time derivative. We illustrate the efficacy of our method through simulations, and we discuss the usefulness of parameters in the controls. Since the control laws are feedback, their values are updated based on data acquired in real time. We also discuss the degradation caused by the delayed data acquisition occurring in practical implementations, and we derive bounds on the delays under which the ISS property is ensured when delays are present.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78067332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove existence of optimal controls for sparse optimal control of a quasilinear elliptic equation in measure spaces and derive first-order necessary optimality conditions. Under additional assumptions also second-order necessary and sufficient optimality conditions are obtained.
{"title":"Sparse optimal control of a quasilinear elliptic PDE in measure spaces","authors":"Fabian Hoppe","doi":"10.3934/mcrf.2022049","DOIUrl":"https://doi.org/10.3934/mcrf.2022049","url":null,"abstract":"<p style='text-indent:20px;'>We prove existence of optimal controls for sparse optimal control of a quasilinear elliptic equation in measure spaces and derive first-order necessary optimality conditions. Under additional assumptions also second-order necessary and sufficient optimality conditions are obtained.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85740249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of estimating (reconstructing) an unknown input for a system of nonlinear differential equations with the Caputo fractional derivative is considered. Information on the position of the system is available for observations and only a part of system's parameters can be measured. The case of measuring all phase coordinates is also presented. The measurements are continuous and the data obtained in them are noisy. The considered problem is ill-posed and, to solve it, we use the method of dynamic inversion. It is based on regularization methods and constructions of positional control theory. In particular, we use the Tikhonov regularization method also known as the smoothing functional method and the Krasovskii extremal aiming method. The approach to estimating an unknown input implies introducing an auxiliary system (a model) with an appropriate rule of forming a control. The proposed estimation algorithm gives approximations of an unknown input and is stable under informational noises and computational errors. As an example illustrating the elaborated technique, a biological model of human immunodeficiency virus disease is used for simulation. The simulation results demonstrate the importance of the approach to on-line estimating unobservable parameters in real processes.
{"title":"Dynamical estimation of a noisy input in a system with a Caputo fractional derivative. The case of continuous measurements of a part of phase coordinates","authors":"P. Surkov","doi":"10.3934/mcrf.2022020","DOIUrl":"https://doi.org/10.3934/mcrf.2022020","url":null,"abstract":"The problem of estimating (reconstructing) an unknown input for a system of nonlinear differential equations with the Caputo fractional derivative is considered. Information on the position of the system is available for observations and only a part of system's parameters can be measured. The case of measuring all phase coordinates is also presented. The measurements are continuous and the data obtained in them are noisy. The considered problem is ill-posed and, to solve it, we use the method of dynamic inversion. It is based on regularization methods and constructions of positional control theory. In particular, we use the Tikhonov regularization method also known as the smoothing functional method and the Krasovskii extremal aiming method. The approach to estimating an unknown input implies introducing an auxiliary system (a model) with an appropriate rule of forming a control. The proposed estimation algorithm gives approximations of an unknown input and is stable under informational noises and computational errors. As an example illustrating the elaborated technique, a biological model of human immunodeficiency virus disease is used for simulation. The simulation results demonstrate the importance of the approach to on-line estimating unobservable parameters in real processes.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81490518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cost evaluation of finite dimensional and approximate null-controllability for the one dimensional half-heat equation","authors":"S. Micu, Constantin Nita","doi":"10.3934/mcrf.2022054","DOIUrl":"https://doi.org/10.3934/mcrf.2022054","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90377408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider a multi-dimensional Bresse with memory-type boundary conditions. By assuming minimal conditions on the resolvent kernel, we establish an optimal explicit and general energy decay result. This result is new and substantially improves earlier results in the literature.
{"title":"New general decay results for a multi-dimensional Bresse system with viscoelastic boundary conditions","authors":"B. Feng, A. Soufyane, M. Afilal","doi":"10.3934/mcrf.2022050","DOIUrl":"https://doi.org/10.3934/mcrf.2022050","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we consider a multi-dimensional Bresse with memory-type boundary conditions. By assuming minimal conditions on the resolvent kernel, we establish an optimal explicit and general energy decay result. This result is new and substantially improves earlier results in the literature.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84804254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with a control constrained optimization problem governed by a nonsmooth elliptic PDE in the presence of a non-differentiable objective. The nonsmooth non-linearity in the state equation is locally Lipschitz continuous and directionally differentiable, while one of the nonsmooth terms appearing in the objective is convex. Since these mappings are not necessarily Gâteaux-differentiable, the application of standard adjoint calculus is excluded. Based on their limited differentiability properties, we derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative.
{"title":"Strong stationarity for a highly nonsmooth optimization problem with control constraints","authors":"Livia M. Betz","doi":"10.3934/mcrf.2022047","DOIUrl":"https://doi.org/10.3934/mcrf.2022047","url":null,"abstract":"This paper deals with a control constrained optimization problem governed by a nonsmooth elliptic PDE in the presence of a non-differentiable objective. The nonsmooth non-linearity in the state equation is locally Lipschitz continuous and directionally differentiable, while one of the nonsmooth terms appearing in the objective is convex. Since these mappings are not necessarily Gâteaux-differentiable, the application of standard adjoint calculus is excluded. Based on their limited differentiability properties, we derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88076723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we consider the nonlinear BBM equation on the torus. We use controls taking values in a finite dimensional space to show that the equation is approximately controllable in begin{document}$ H^1(mathbb{T}) $end{document}. We also show that the equation is not exactly controllable in begin{document}$ H^s(mathbb{T}) $end{document} for begin{document}$ sin[1,2[ $end{document}.
In this article, we consider the nonlinear BBM equation on the torus. We use controls taking values in a finite dimensional space to show that the equation is approximately controllable in begin{document}$ H^1(mathbb{T}) $end{document}. We also show that the equation is not exactly controllable in begin{document}$ H^s(mathbb{T}) $end{document} for begin{document}$ sin[1,2[ $end{document}.
{"title":"On the controllability of the BBM equation","authors":"Melek Jellouli","doi":"10.3934/mcrf.2022002","DOIUrl":"https://doi.org/10.3934/mcrf.2022002","url":null,"abstract":"<p style='text-indent:20px;'>In this article, we consider the nonlinear BBM equation on the torus. We use controls taking values in a finite dimensional space to show that the equation is approximately controllable in <inline-formula><tex-math id=\"M1\">begin{document}$ H^1(mathbb{T}) $end{document}</tex-math></inline-formula>. We also show that the equation is not exactly controllable in <inline-formula><tex-math id=\"M2\">begin{document}$ H^s(mathbb{T}) $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M3\">begin{document}$ sin[1,2[ $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82273502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.
{"title":"The application of a universal separating vector lemma to optimal sampled-data control problems with nonsmooth Mayer cost function","authors":"S. Adly, L. Bourdin, Gaurav Dhar","doi":"10.3934/mcrf.2022039","DOIUrl":"https://doi.org/10.3934/mcrf.2022039","url":null,"abstract":"In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74993689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Duality between controllability and observability of stochastic linear evolution equations with general filtration","authors":"Lu Lin","doi":"10.3934/mcrf.2022056","DOIUrl":"https://doi.org/10.3934/mcrf.2022056","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72753280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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