{"title":"最小假设下耦合扫瞄过程的最优控制","authors":"Samara Chamoun, Vera Zeidan","doi":"10.1007/s00245-025-10268-0","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, the study of nonsmooth optimal control problems (<i>P</i>) involving a controlled sweeping process with <i>three</i> main characteristics is launched. First, the sweeping sets are <i>nonsmooth</i>, <i>time-dependent</i>, and uniformly prox-regular. Second, the sweeping process is <i>coupled</i> with a controlled differential equation. Third, a <i>joint</i>-state endpoints constraint set <i>S</i> is present. This general model incorporates different important controlled submodels, such as a class of second order sweeping processes, and coupled evolution variational inequalities. A full form of the <i>nonsmooth</i> Pontryagin maximum principle for <i>strong</i> local minimizers in (<i>P</i>) is derived for <i>bounded or unbounded moving</i> sweeping sets satisfying <i>local</i> constraint qualifications (CQ) <i>without</i> any additional restriction. The existence and uniqueness of a Lipschitz solution for the Cauchy problem of our dynamic is established and the existence of an optimal solution for (<i>P</i>) is obtained. Two of the novelties in achieving the first goal are (i) the construction of a problem over <i>truncated</i> sweeping sets and <i>truncated</i> joint endpoints constraint set that has the same strong local minimizer as (<i>P</i>) and its (CQ) automatically holds, and (ii) the <i>complete redesign</i> of the exponential-penalty approximation technique for problems with moving sweeping sets that <i>do not require</i> any special assumption on the sets, their corners, or on the gradients of their generators. The utility of the optimality conditions is illustrated with an example.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"92 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10268-0.pdf","citationCount":"0","resultStr":"{\"title\":\"Optimal Control for Coupled Sweeping Processes Under Minimal Assumptions\",\"authors\":\"Samara Chamoun, Vera Zeidan\",\"doi\":\"10.1007/s00245-025-10268-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, the study of nonsmooth optimal control problems (<i>P</i>) involving a controlled sweeping process with <i>three</i> main characteristics is launched. First, the sweeping sets are <i>nonsmooth</i>, <i>time-dependent</i>, and uniformly prox-regular. Second, the sweeping process is <i>coupled</i> with a controlled differential equation. Third, a <i>joint</i>-state endpoints constraint set <i>S</i> is present. This general model incorporates different important controlled submodels, such as a class of second order sweeping processes, and coupled evolution variational inequalities. A full form of the <i>nonsmooth</i> Pontryagin maximum principle for <i>strong</i> local minimizers in (<i>P</i>) is derived for <i>bounded or unbounded moving</i> sweeping sets satisfying <i>local</i> constraint qualifications (CQ) <i>without</i> any additional restriction. The existence and uniqueness of a Lipschitz solution for the Cauchy problem of our dynamic is established and the existence of an optimal solution for (<i>P</i>) is obtained. Two of the novelties in achieving the first goal are (i) the construction of a problem over <i>truncated</i> sweeping sets and <i>truncated</i> joint endpoints constraint set that has the same strong local minimizer as (<i>P</i>) and its (CQ) automatically holds, and (ii) the <i>complete redesign</i> of the exponential-penalty approximation technique for problems with moving sweeping sets that <i>do not require</i> any special assumption on the sets, their corners, or on the gradients of their generators. 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Optimal Control for Coupled Sweeping Processes Under Minimal Assumptions
In this paper, the study of nonsmooth optimal control problems (P) involving a controlled sweeping process with three main characteristics is launched. First, the sweeping sets are nonsmooth, time-dependent, and uniformly prox-regular. Second, the sweeping process is coupled with a controlled differential equation. Third, a joint-state endpoints constraint set S is present. This general model incorporates different important controlled submodels, such as a class of second order sweeping processes, and coupled evolution variational inequalities. A full form of the nonsmooth Pontryagin maximum principle for strong local minimizers in (P) is derived for bounded or unbounded moving sweeping sets satisfying local constraint qualifications (CQ) without any additional restriction. The existence and uniqueness of a Lipschitz solution for the Cauchy problem of our dynamic is established and the existence of an optimal solution for (P) is obtained. Two of the novelties in achieving the first goal are (i) the construction of a problem over truncated sweeping sets and truncated joint endpoints constraint set that has the same strong local minimizer as (P) and its (CQ) automatically holds, and (ii) the complete redesign of the exponential-penalty approximation technique for problems with moving sweeping sets that do not require any special assumption on the sets, their corners, or on the gradients of their generators. The utility of the optimality conditions is illustrated with an example.
期刊介绍:
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.