Mathematical Control and Related Fields最新文献

{"title":"Stochastic linear-quadratic control with a jump and regime switching on a random horizon","authors":"Ying Hu, Xiaomin Shi, Z. Xu","doi":"10.3934/mcrf.2022051","DOIUrl":"https://doi.org/10.3934/mcrf.2022051","url":null,"abstract":"In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a horizon [0, T ∧ τ ], where τ is a given random jump time for the underlying state process and T is a constant. We obtain an explicit optimal state feedback control and explicit optimal cost value by solving a system of stochastic Riccati equations (SREs) with jumps on [0, T ∧ τ ]. By the decomposition approach stemming from filtration enlargement theory, we express the solution of the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon [0, T ]. Solving the latter system is the key theoretical contribution of this paper and we establish this for three different cases, one of which seems to be new in the literature. These results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"25 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82583841","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":"Feedback law to stabilize linear infinite-dimensional systems","authors":"Yaxing Ma, GengshengBB Wang, Huaiqiang Yu","doi":"10.3934/mcrf.2022031","DOIUrl":"https://doi.org/10.3934/mcrf.2022031","url":null,"abstract":"<p style='text-indent:20px;'>We design a new feedback law to stabilize the linear infinite-dimensional control system, where the state operator generates a <inline-formula><tex-math id=\"M1\">begin{document}$ C_0 $end{document}</tex-math></inline-formula>-group, and the control operator is unbounded. Our feedback law is based on the integration of a mutated Gramian operator-valued function. In the structure of the aforementioned mutated Gramian operator, we utilize the weak observability inequality in [<xref ref-type=\"bibr\" rid=\"b21\">21</xref>,<xref ref-type=\"bibr\" rid=\"b13\">13</xref>] and borrow some idea used to construct generalized Gramian operators in [<xref ref-type=\"bibr\" rid=\"b11\">11</xref>,<xref ref-type=\"bibr\" rid=\"b23\">23</xref>,<xref ref-type=\"bibr\" rid=\"b24\">24</xref>]. Unlike most related works where the exact controllability is required, we only assume the above-mentioned weak observability inequality, which is equivalent to the stabilizability of the system.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"41 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84543917","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":"Non-homogeneous stochastic LQ control with regime switching and random coefficients","authors":"Ying Hu, Xiaomin Shi, Z. Xu","doi":"10.3934/mcrf.2023021","DOIUrl":"https://doi.org/10.3934/mcrf.2023021","url":null,"abstract":"This paper is concerned with a general non-homogeneous stochastic linear quadratic (LQ) control problem with regime switching and random coefficients. We obtain the explicit optimal state feedback control and optimal value for this problem in terms of two systems of backward stochastic differential equations (BSDEs): one is the famous stochastic Riccati equation and the other one is a new linear multi-dimensional BSDE with all coefficients being unbounded. The existence and uniqueness of the solutions to these two systems of BSDEs are proved by means of BMO martingales and contraction mapping method. At last, the theory is applied to study an asset-liability management problem under the mean-variance criteria.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83125645","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":"Controllability of the linear elasticity as a first-order system using a stabilized space-time mixed formulation","authors":"Arthur Bottois, N. Cîndea","doi":"10.3934/mcrf.2022028","DOIUrl":"https://doi.org/10.3934/mcrf.2022028","url":null,"abstract":"<p style='text-indent:20px;'>The aim of this paper is to study the boundary controllability of the linear elasticity system as a first-order system in both space and time. Using the observability inequality known for the usual second-order elasticity system, we deduce an equivalent observability inequality for the associated first-order system. Then, the control of minimal <inline-formula><tex-math id=\"M1\">begin{document}$ L^2 $end{document}</tex-math></inline-formula>-norm can be found as the solution to a space-time mixed formulation. This first-order framework is particularly interesting from a numerical perspective since it is possible to solve the space-time mixed formulation using only piecewise linear <inline-formula><tex-math id=\"M2\">begin{document}$ C^0 $end{document}</tex-math></inline-formula>-finite elements. Numerical simulations illustrate the theoretical results.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"11 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82524862","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":"Input-to-state stability of non-autonomous infinite-dimensional control systems","authors":"H. Damak","doi":"10.3934/mcrf.2022035","DOIUrl":"https://doi.org/10.3934/mcrf.2022035","url":null,"abstract":"This paper addresses input-to-state stability (ISS) and integral input-to-state stability (iISS) for non-autonomous infinite-dimensional control systems. With the notion of uniformly exponential stability scalar function, ISS and iISS are considered based on indefinite Lyapunov functions. In addition, we obtain several necessary and sufficient characterizations of the iISS property, expressed in terms of dissipation inequalities. As a result, the iISS criteria of non-autonomous bilinear systems is also established. Furthermore, an illustrative example is given to show the applicability of the results.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"50 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75800238","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":"Exponential and polynomial stabilization of laminated beams with two history memories","authors":"Teófanes Quispe Méndez, V. Cabanillas, B. Feng","doi":"10.3934/mcrf.2022037","DOIUrl":"https://doi.org/10.3934/mcrf.2022037","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, a laminated beam system with two history-type controls is studied. One of the controls acts on the effective rotation angle and the other on the slip equation. The latter control replaces the structural damping usually considered in this model in the literature. Using the semigroup of linear operators approach, we prove the system is globally well-posed. We establish the exponential stability of the system provided the equal-speed waves propagation holds. Otherwise, the system lacks exponential stability. The polynomial decay with rate <inline-formula><tex-math id=\"M1\">begin{document}$ t^{-frac{1}{4}} $end{document}</tex-math></inline-formula> is also proved using the theorem due to Borichev and Tomilov.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"48 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85387668","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":"On optimal control problem for the Perona-Malik equation and its approximation","authors":"Yaroslav Kohut, O. Kupenko","doi":"10.3934/mcrf.2022045","DOIUrl":"https://doi.org/10.3934/mcrf.2022045","url":null,"abstract":"<p style='text-indent:20px;'>We discuss the existence of solutions to an optimal control problem for the Neumann boundary value problem for the Perona-Malik equations. The control variable <inline-formula><tex-math id=\"M5\">begin{document}$ v $end{document}</tex-math></inline-formula> is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution <inline-formula><tex-math id=\"M6\">begin{document}$ u_din L^2(Omega) $end{document}</tex-math></inline-formula> and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and derive some optimality conditions for the approximating problems.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"4 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78754720","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":"Expected power utility maximization with delay for insurers under the 4/2 stochastic volatility model","authors":"H. Hata, Kazuhiro Yasuda","doi":"10.3934/mcrf.2022055","DOIUrl":"https://doi.org/10.3934/mcrf.2022055","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"3 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90160789","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":"Control of reaction-diffusion models in biology and social sciences","authors":"Domènec Ruiz-Balet, E. Zuazua","doi":"10.3934/mcrf.2022032","DOIUrl":"https://doi.org/10.3934/mcrf.2022032","url":null,"abstract":"<p style='text-indent:20px;'>These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.</p><p style='text-indent:20px;'>The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.</p><p style='text-indent:20px;'>We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.</p><p style='text-indent:20px;'>The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.</p><p style='text-indent:20px;'>We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems' dynamics and, in particular, the use of traveling waves and steady-states' paths, can be employed to build controls driving the system to the desired target.</p><p style='text-indent:20px;'>We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.</p><p style='text-indent:20px;'>Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"216 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75792051","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":"Correction To \"Null controllability of a nonlinear age, space and two-sex structured population dynamics model\"","authors":"Y. Simporé, O. Traore","doi":"10.3934/mcrf.2022033","DOIUrl":"https://doi.org/10.3934/mcrf.2022033","url":null,"abstract":"<jats:p xml:lang=\"fr\" />","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"90 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79409344","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}