Minimum Time for a Hybrid System with Thermostatic Switchings

Objective function design for robust optimality of linear control under state-constraints and uncertainty

“…To this end, as explained next, it seems necessary to introduce a discontinuity (switching term) in the objective function g, and this fact may lead to some problem for the uniqueness of the Hamilton-Jacobi equations. To overcome such a difficulty, we approximate that switching term by a delayed thermostat and then study a suitable kind of hybrid system, in the spirit of Bagagiolo [1] (see Fig. 7).…”

Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics

“…We refer the reader to the book by Bardi and Capuzzo Dolcetta [8] for the theory of viscosity solutions and applications to optimal control problems. Other thermostatic problems of this type, but with only one switching variable w ∈ {−1, 1}, are studied, with similar techniques, for instance in Bagagiolo [4]. In the present case, the presence of more than one thermostatic switching variables leads to some new problems to be treated, such as the discontinuity of the boundary data and the more complex geometry of the branches.…”

A Differential Game with Exit Costs

“…Hence, the switching evolution of the parameter w is not directly at disposal of the external controller, but it follows some internal switching rules which are intrinsic to the system. In [1,3], the value function is proven to be the unique viscosity solution of a suitably coupled system of HJB equations, where the coupling is given by the boundary conditions in the regions where the thermostat certainly assumes a constant value (cannot switch). This is done by splitting the optimal control problem in some problems of exit time kind: in every space-region where the thermostat is constant, the problem is equivalent to an exit-time problem with unknown exit-cost given by the value function itself evaluated in the other region of constancy for w. Then, an ad hoc fixed point procedure is applied.…”

“…In [1,3,4] some motivations and applications for studying optimal control problems with thermostatic dynamics are given. Similar motivations certainly suggest the study of differential games with thermostatic dynamics.…”

“…In the case of non-anticipating strategies, a similar construction as in [21] is forbidden. In the present paper, using the fact that the dynamics of the two players are decoupled with respect to the space-variables and to the controls (see (1)), and also adding a decoupled feature in the controls for the cost (see (48)), we are able to suitably adapt Soner's construction in order to get the desired state-constraint nonanticipating strategy. Such assumptions on decoupled dynamics, at the present moment, seems almost necessary in order to get this kind of results.…”