From the theory of nonlinear optimal control problems it is known that the solution stability w.r.t. data perturbations and conditions for strict local optimality are closely related facts. For important classes of control problems, sufficient optimality conditions can be formulated as a combination of the independence of active constraints' gradients and certain coercivity criteria. In the case of discontinuous controls, however, common pointwise coercivity approaches may fail. In the paper, we consider sufficient optimality conditions for strong local minimizers which make use of an integrated Hamilton-Jacobi inequality. In the case of linear system dynamics, we show that the solution stability (including the switching points localization) is ensured under relatively mild regularity assumptions on the switching function zeros. For the objective functional, local quadratic growth estimates in L(1) sense are provided. An example illustrates stability as well as instability effects in case the regularity condition is violated.