An iterative procedure for constructing minimax and viscosity solutions to the Hamilton-Jacobi equations and its generalization. (English. Russian original) Zbl 0974.35022

Proc. Steklov Inst. Math. 224, 286-309 (1999); translation from Tr. Mat. Inst. Steklova 224, 311-334 (1999).
This article is an extended version of the authors’ note in [Dokl. Math. 53, No. 3, 416-419 (1996; Zbl 0902.35023)] concerning a certain iterative-type characterization of the so-called minimax solutions in [A. I. Subbotin, Generalized solutions of first-order PDEs. The dynamical optimization perspective, Basel (1994; Zbl 0820.35003)] of problems of the form: \[ u_t + H(t,x,u,D_xu)=0, \quad (t,x)\in G:=(0,\theta)\times \mathbb{R}^n, \quad u(\theta ,x)= \sigma (x). \] Under certain hypotheses on the Hamiltonian \(H\) one associates the differential inclusions \[ (x',z')\in E_p(x,z):=\{(u,v); \;||u||\leq \lambda (x), \quad v=\langle u,p\rangle-H(t,x,z,p)\} , \quad p\in \mathbb{R}^n \] and one defines a minimax solution as a continuous function \(u:C(G)\to \mathbb{R}\) that satisfies the boundary condition and whose graph is weakly invariant (“viable”) with respect to all these differential inclusions.
Next, for \((t_0,x_0,z_0)\in D:=[0,\theta]\times \mathbb{R}^n\times \mathbb{R}\) one denotes by \(S_p(t_0,x_0,z_0)\) the set of all \(AC\) solutions on the interval \([0,\theta]\) satisfying: \(x(t_0)=x_0\), \(z(t_0)=z_0\) and one introduces the operators: \[ \Pi(M):=\{(t_0,x_0,z_0); \;\forall p\in \mathbb{R}^n\;\exists (x,z)\in S_p (t_0,x_0,z_0): (x(\theta),z(\theta))\in M \},\;M\subset \mathbb{R}^n\times \mathbb{R}, \]
\[ A(W):= \{(t_0,x_0,z_0); \;\forall p\in \mathbb{R}^n \;\exists (x,z)\in S_p (t_0,x_0,z_0), \;\tau \in [t_0,\theta]: \;(\tau,x(\tau),z(\tau))\in W \}, \] \(W\subset D\). The main result of the paper seems to be Theorem 1.1 stating that a minimax solution exists, its epigraph \(U\) and, respectively, the hypograph \(V\) being given by: \[ U= \lim_{i\to \infty}U_i= \bigcap_{i= 0}^\infty U_i, \quad V= \lim_{i\to \infty}V_i= \bigcap_{i= 0}^\infty V_i \] where \(U_i\), \(V_i\) are recurrently defined by: \[ U_0:=\Pi(\text{epi}(\sigma)), \quad U_{i+1}:=A(U_i), \quad V_0:=\Pi(\text{hypo} (\sigma)), \quad V_{i+1}:=A(V_i). \] In the second part of the paper the authors present an abstract version of this iterative procedure in terms of some types of “monotonic operators” on spaces of sets.
For the entire collection see [Zbl 0942.00074].


35F20 Nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
34A60 Ordinary differential inclusions