Maximum under continuous-discrete-time dynamic with target and viability constraints

The viable maximum of integral(T)(0) L(x(t), u(t))dt of a continuous function L is an element of L-1 (IR2m+1, IR+) under a dynamic x'(t) is an element of F(x(t)) under constraint x(t) is an element of K where K is closed is obtained on the boundary of the capture-viability kernel in the direction of high y of the target K x {0} viable in K x IR+ under the extended dynamic (x'(t), y'(t)) is an element of (F(x(t)), - L(x(t), u(t))). The result holds true with discrete- continuous-time measurable controls. Example and application to hybrid dynamics under viability constraints and target are given.