A Bellman approach for two-domains optimal control problems in \(\mathbb{R}^N\) (Q2842250)

In this interesting paper, the authors study the problem of defining in a right way the dynamics, the cost and the Bellman equation for an optimal control problem whose dynamics and costs are independently (and differently) given in two half-spaces of \(\mathbb{R}^N\).NEWLINENEWLINEIn particular they consider the hyperplane \({\mathcal H}=\{x_N=0\}\), the two half-spaces \(\Omega_1=\{x_N>0\},\Omega_2=\{x_N<0\}\), two controlled dynamics \(b_i:\Omega_i\times A_i\to\mathbb{R}^N\), two running costs \(\ell_i:\Omega_i\times A_i\to\mathbb{R}^N\), and the two corresponding Hamilton-Jacobi-Bellman equations \(H_i(x,u,Du)=0\), \(i=1,2\). By convexification, they suitably define a controlled dynamics on \({\mathcal H}\), and study the following problem:NEWLINENEWLINENEWLINE\[NEWLINE \begin{cases} H_1(x,u,Du)=0&\text{in } \Omega_1,\\ H_2(x,u,Du)=0&\text{in } \Omega_2,\\ \min\{H_1(x,u,Du),H_2(x,u,Du)\}\leq0&\text{on } {\mathcal H}\\ \displaystyle \max\{H_1(x,u,Du),H_2(x,u,Du)\}\geq0&\text{on } {\mathcal H}. \end{cases} NEWLINE\]NEWLINENEWLINENEWLINEThey prove an ad-hoc comparison result for viscosity super- and subsolutions (here the usual double varying technique does not hold). Then they prove that two suitably defined value functions are the minimal viscosity supersolution and the maximal viscosity subsolution of the problem. They also give some further properties of one of these two solutions on \(\mathcal H\). Finally, in the one dimensional setting, they consider two possible approximations/relaxations of the problem.NEWLINENEWLINEThe problem studied here is rather new and with many interesting possible applications, think for instance to the control of hybrid systems.NEWLINENEWLINEThe reviewer would like to point out that another possible approximation/relaxation of the problem can be given by a so-called delayed thermostatic approximation of the switching across the hyperplane. See for instance the following papers (and the references therein):NEWLINENEWLINE[\textit{F. Bagagiolo}, Discrete Contin. Dyn. Syst., Ser. B 5, No. 2, 239--264 (2005; Zbl 1120.49021)],NEWLINENEWLINE [\textit{F. Bagagiolo} and \textit{K. Danieli}, Nonlinear Anal., Hybrid Syst. 6, No. 2, 824--838 (2012; Zbl 1242.49057)].