Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems

Abstract

This article deals with the Lipschitz regularity of the “approximate” minimizers for the Bolza type control functional of the form \[ J t ( y , u ) ≔ ∫ t T Λ ( s , y ( s ) , u ( s ) ) d s + g ( y ( T ) ) J_t(y,u)≔\int _t^T\Lambda (s,y(s), u(s))\,ds+g(y(T)) \] among the pairs ( y , u ) (y,u) satisfying a prescribed initial condition y ( t ) = x y(t)=x , where the state y y is absolutely continuous, the control u u is summable and the dynamic is controlled-linear of the form y ′ = b ( y ) u y’=b(y)u . For b ≡ 1 b\equiv 1 the above becomes a problem of the calculus of variations. The Lagrangian Λ ( s , y , u ) \Lambda (s,y,u) is assumed to be either convex in the variable u u on every half-line from the origin (radial convexity in u u ), or partial differentiable in the control variable and satisfies a local Lipschitz regularity on the time variable, named Condition (S). It is allowed to be extended valued, discontinuous in y y or in u u , and non convex in u u . We assume a very mild growth condition, actually a violation of the Du Bois-Reymond–Erdmann equation for high values of the control, that is fulfilled if the Lagrangian is coercive as well as in some almost linear cases. The main result states that, given any admissible pair ( y , u ) (y,u) , there exists a more convenient admissible pair ( y ¯ , u ¯ ) (\overline y, \overline u) for J t J_t where u ¯ \overline u is bounded, y ¯ \overline y is Lipschitz, with bounds and ranks that are uniform with respect to 查看原文

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