A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls,...

We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power ${(-A)}^{\beta }$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already...

In this note, we verify the conjecture of Barron, Evans and Jensen [3] regarding the characterization of viscosity solutions of general Aronsson equations in terms of the properties of associated forward and backwards Hamilton-Jacobi flows. A special case of this result is analogous to the characterization of infinity harmonic functions in terms of convexity and concavity of the functions $r\mapsto {max}_{y\in B_r\left(x\right)}u\left(y\right)$ and $r\mapsto {min}_{y\in B_r\left(x\right)}u\left(y\right)$, respectively.

This article is devoted to the optimal control of state equations with memory of the form: $\dot{x}\left(t\right)=F(x\left(t\right),u\left(t\right),{\int }_0^{+\infty }A\left(s\right)x(t-s)\mathrm{d}s),t>0,$with initial conditions $x\left(0\right)=x,x(-s)=z\left(s\right),s>0$. Denoting by $y_{x,z,u}$ the solution of the previous Cauchy problem and:$v(x,z):={inf}_{u\in V}\left\{{\int }_0^{+\infty }{\mathrm{e}}^{-\lambda s}L(y_{x,z,u}\left(s\right),u\left(s\right))\mathrm{d}s\right\}$ where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form:$\lambda v(x,z)+H(x,z,{\nabla }_{x}v(x,z))+\langle D_{z}v(x,z),\dot{z}\rangle =0$ in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

MSC 2010: 26A33, 70H25, 46F12, 34K37 Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and the order of the derivation. The Euler–Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation....