Extended Euler–Lagrange and Hamiltonian Conditions in Optimal Control of Sweeping Processes with Controlled Moving Sets
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This paper concerns optimal control problems for a class of sweeping processes governed by discontinuous unbounded differential inclusions that are described via normal cone mappings to controlled moving sets. Largely motivated by applications to hysteresis, we consider a general setting where moving sets are given as inverse images of closed subsets of finite-dimensional spaces under nonlinear differentiable mappings dependent on both state and control variables. Developing the method of discrete approximations and employing generalized differential tools of first-order and second-order variational analysis allow us to derive nondegenerate necessary optimality conditions for such problems in extended Euler–Lagrange and Hamiltonian forms involving the Hamiltonian maximization. The latter conditions of the Pontryagin Maximum Principle type are the first in the literature for optimal control of sweeping processes with control-dependent moving sets.
KeywordsOptimal control Sweeping process Variational analysis Discrete approximations Generalized differentiation Euler–Lagrange and Hamiltonian formalisms Maximum principle Rate-independent operators
Mathematics Subject Classification49J52 49J53 49K24 49M25 90C30
Research of the second author was partly supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, and by the USA Air Force Office of Scientific Research under Grant #15RT0462.
- 1.Moreau, J.J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.) Proceedings from CIME New Variational Techniques in Mathematical Physics, pp. 173–322. Cremonese, Rome (1974)Google Scholar
- 2.Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis, pp. 99–182. International Press, Boston (2010)Google Scholar
- 17.de Pinho, M.D.R., Ferreira, M.M.A., Smirnov, G.V.: Optimal control involving sweeping processes. Set-Valued Var. Anal. (to appear)Google Scholar