2024/08/05 by Abderrahim Bouach, Tahar Haddad, Bouach, Abderrahim +3
Engineering · Mathematics · #49J53 #49K22 #49K24 #94C99 #Applied mathematics #Approximations of π #Differential (mechanical device) #Differential inclusion #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical optimization #Mathematics #Numerical methods for differential equations #Optimization and Control (math.OC) #Physics #Thermodynamics #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.2408.02856
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2024/08/05 · openalex created_date 2024/10/22 · openalex updated_date 2026/08/06
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 W1,2-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.