2013/09/15 by Vladimir Gaitsgory, Gaitsgory, Vladimir, Sergei Rossomakhine +1
Computer Science · Mathematics · #34A60 #34C29 #34E15 #93C70 #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Differential Equations Analysis #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1309.3734
openalex publication_date 2013/09/15 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
The paper aims at the development of an apparatus for analysis and\nconstruction of near optimal solutions of singularly perturbed (SP) optimal\ncontrols problems (that is, problems of optimal control of SP systems)\nconsidered on the infinite time horizon.\n We mostly focus on problems with time discounting criteria but a possibility\nof the extension of results to periodic optimization problems is discussed as\nwell. Our consideration is based on earlier results on averaging of SP control\nsystems and on linear programming formulations of optimal control problems. The\nidea that we exploit is to first asymptotically approximate a given problem of\noptimal control of the SP system by a certain averaged optimal control problem,\nthen reformulate this averaged problem as an infinite-dimensional (ID) linear\nprogramming (LP) problem, and then approximate the latter by semi-infinite LP\nproblems. We show that the optimal solution of these semi-infinite LP problems\nand their duals (that can be found with the help of a modification of an\navailable LP software) allow one to construct near optimal controls of the SP\nsystem. We demonstrate the construction with two numerical examples.\n