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

This article deals with the Lipschitz regularity of the ''approximate`` minimizers for the Bolza type control functional of the form $$J_t(y,u):=\int_t^T\Lambda(s,y(s), u(s))\,ds+g(y(T))$$ among the pairs $(y,u)$ satisfying a prescribed initial condition $y(t)=x$, where the state $y$ is absolutely continuous, the control $u$ is summable and the dynamic is controlled-linear of the form $y'=b(y)u$. For $b\equiv 1$ the above becomes a problem of the calculus of variations. The Lagrangian $\Lambda(s,y,u)$ is assumed to be either convex in the variable $u$ on every half-line from the origin (radial convexity in $u$), 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 $y$ or in $u$, and non convex in $u$.\\ We assume a very mild growth condition, 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 $(y,u)$, there exists a more convenient admissible pair $(\overline y, \overline u)$ for $J_t$ where $\overline u$ is bounded, $\overline y$ is Lipschitz, with bounds and ranks that are uniform with respect to $t,x$ in the compact subsets of $[0,T[\times\mathbb R^n$. The result is new even in the superlinear case. As a consequence, there are minimizing sequences that are formed by pairs of equi-Lipschitz trajectories and equi--bounded controls.\\ A new existence and regularity result follows without assuming any kind of Lipschitzianity in the state variable.\\ We deduce, without any need of growth conditions, the nonoccurrence of the Lavrentiev phenomenon for a wide class of Lagrangians containing those that satisfy Condition (S), are bounded on bounded sets ``well'' inside the effective domain and are radially convex in the control variable.