State constrained control problems with neither coercivity nor L1 bounds on the controls

A state-constrained, nonlinear, minimum problem is considered with dynamics depending sublinearly on a control which is not bounded in theL 1 norm. Because of the lack of coercivity, the value map fails to be continuous, even in the unconstrained case. However, we prove that under suitable assumptions—which guarantee the continuity of the value maps of the problems withL 1-bounded controls—the value map is upper semicontinuous and solves a Bellman equation with a continuous Hamiltonian. Moreover, the map obtained by replacing its values at the horizon t=T with the values of the cost function turns out to be the maximal subsolution of the corresponding value problem.