Discrete approximations and optimality conditions for integro-differential inclusions
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| Format: | Preprint |
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2024
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| _version_ | 1866912058034880512 |
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| author | Bouach, Abderrahim Haddad, Tahar Mordukhovich, Boris S. |
| author_facet | Bouach, Abderrahim Haddad, Tahar Mordukhovich, Boris S. |
| contents | This paper addresses a new class of generalized Bolza problems governed by nonconvex integro-differential inclusions with endpoint constraints on trajectories, where the integral terms are given in the general (with time-dependent integrands in the dynamics) Volterra form. We pursue here a threefold goal. First we construct well-posed approximations of continuous-time integro-differential systems by their discrete-time counterparts with showing that any feasible solution to the original system can be strongly approximated in the $W^{1,2}$-norm topology by piecewise-linear extensions of feasible discrete trajectories. This allows us to verify in turn the strong convergence of discrete optimal solutions to a prescribed local minimizer for the original problem. Facing intrinsic nonsmoothness of original integro-differential problem and its discrete approximations, we employ appropriate tools of generalized differentiation in variational analysis to derive necessary optimality conditions for discrete-time problems (which is our second goal) and finally accomplish our third goal to obtain necessary conditions for the original continuous-time problems by passing to the limit from discrete approximations. In this way we establish, in particular, a novel necessary optimality condition of the Volterra type, which is the crucial result for dynamic optimization of integro-differential inclusions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete approximations and optimality conditions for integro-differential inclusions Bouach, Abderrahim Haddad, Tahar Mordukhovich, Boris S. Optimization and Control 49K24, 49K22, 49J53, 94C99 This paper addresses a new class of generalized Bolza problems governed by nonconvex integro-differential inclusions with endpoint constraints on trajectories, where the integral terms are given in the general (with time-dependent integrands in the dynamics) Volterra form. We pursue here a threefold goal. First we construct well-posed approximations of continuous-time integro-differential systems by their discrete-time counterparts with showing that any feasible solution to the original system can be strongly approximated in the $W^{1,2}$-norm topology by piecewise-linear extensions of feasible discrete trajectories. This allows us to verify in turn the strong convergence of discrete optimal solutions to a prescribed local minimizer for the original problem. Facing intrinsic nonsmoothness of original integro-differential problem and its discrete approximations, we employ appropriate tools of generalized differentiation in variational analysis to derive necessary optimality conditions for discrete-time problems (which is our second goal) and finally accomplish our third goal to obtain necessary conditions for the original continuous-time problems by passing to the limit from discrete approximations. In this way we establish, in particular, a novel necessary optimality condition of the Volterra type, which is the crucial result for dynamic optimization of integro-differential inclusions. |
| title | Discrete approximations and optimality conditions for integro-differential inclusions |
| topic | Optimization and Control 49K24, 49K22, 49J53, 94C99 |
| url | https://arxiv.org/abs/2408.02856 |