Continuity properties of the cheap control problem without stability

An open problem concerning the convergence of the optimal cost for the cheap control problem without stability is solved. It turns out that the definition of a new type of linear-quadratic optimal control problems is necessary. This new problem requires the infimization of the cost functional under the constraint that the state trajectory modulo a certain subspace vanishes as time goes to infinity. The associated optimal cost turns out to be the limit of the optimal cost for the cheap control problem without stability. Moreover, for left invertible systems, the optimal control, state and output for the perturbed problem tend to the optimal control, state and output for the new problem. Also a characterization of the optimal cost for the latter problem is given in terms of the dissipation inequality.