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":"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":"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":"Second-order problems involving time-dependent subdifferential operators and application to control","authors":"S. Saïdi, Fatima Fennour","doi":"10.3934/mcrf.2022019","DOIUrl":"https://doi.org/10.3934/mcrf.2022019","url":null,"abstract":"The paper provides a new result concerning the existence of solutions for second-order evolution problems associated with time-dependent subdifferential operators involving both single-valued and mixed semi-continuous set-valued perturbations. Optimal control problems corresponding to such differential inclusions using relaxation theorems with Young measures are investigated. The existence of solutions for a coupled system governed by a second-order differential equation with an evolution problem is also addressed.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"82 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":"82884428","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":"Optimal reinsurance-investment problem for a general insurance company under a generalized dynamic contagion claim model","authors":"Fan Wu, Xin Zhang, Zhibin Liang","doi":"10.3934/mcrf.2022030","DOIUrl":"https://doi.org/10.3934/mcrf.2022030","url":null,"abstract":"In this paper, we study an optimal management problem for a general insurance company which holds shares of an insurance company and a reinsurance company. The general company aims to derive the equilibrium reinsurance-investment strategy under the mean-variance criterion. The claim process described by a generalized compound dynamic contagion process introduced by [18] which allows for self-exciting and externally-exciting clustering effect for the claim arrivals and the processes of the risky assets are described by the jump-diffusion models. Based on practical considerations, we suppose that the externally-exciting clustering effect will simultaneously affect both the price of risky assets and the intensity of claims. To overcome the inconsistency issue caused by the mean-variance criterion, we formulate the optimization problem as an embedded game and solve it via a corresponding extended Hamilton-Jacobi-Bellman equation. The equilibrium reinsurance-investment strategy is obtained, which depends on a solution to an ordinary differential equation. In addition, we demonstrate the derived equilibrium strategy and the economic implications behind it through a large number of mathematical analysis and numerical examples.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"67 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86873121","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":"Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates","authors":"Q. Tran, Harbir Antil, Hugo S Díaz","doi":"10.3934/mcrf.2022003","DOIUrl":"https://doi.org/10.3934/mcrf.2022003","url":null,"abstract":"We consider an optimal control problem governed by parameterized stationary Maxwell's system with the Gauss's law. The parameters enter through dielectric, magnetic permeability, and charge density. Moreover, the parameter set is assumed to be compact. We discretize the electric field by a finite element method and use variational discretization concept for the control. We present a reduced basis method for the optimal control problem and establish the uniform convergence of the reduced order solutions to that of the original full-dimensional problem provided that the snapshot parameter sample is dense in the parameter set, with an appropriate parameter separability rule. Finally, we establish the absolute a posteriori error estimator for the reduced order solutions and the corresponding cost functions in terms of the state and adjoint residuals.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"84 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86855076","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":"The second-order maximum principle for partially observed optimal controls","authors":"Mengzhen Li, Zhanghua Wu","doi":"10.3934/mcrf.2022059","DOIUrl":"https://doi.org/10.3934/mcrf.2022059","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"54 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84596324","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}