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We discuss a general procedure for arriving at the Hamilton-Jacobi equation of second-class constrained systems, and illustrate it in terms of a number of examples by explicitely obtaining the respective Hamilton principal function, and…

In this paper we introduce a notion of viscosity solutions for Eikonal equations defined on topological networks. Existence of a solution for the Dirichlet problem is obtained via representation formulas involving a distance function…

This paper is devoted to the stochastic optimal control problem of infinite-dimensional differential systems allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases studied by…

This article establishes a stochastic homogenization result for the first order Hamilton-Jacobi equation on a Riemannian manifold $M$, in the context of a stationary ergodic random environment. The setting involves a finitely generated…

This paper is concerned with geometric motion of a closed surface whose velocity depends on a nonlocal quantity of the enclosed region. Using the level set formulation, we study a class of nonlocal Hamilton--Jacobi equations and establish a…

We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}_{k \in \mathbb{N}}$ in $\mathbb{R}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \mathbb{N}$. We obtain rates of convergence of…

We present a new formulation for the computation of solutions of a class of Hamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of co-dimension one. For the class of equations considered in this paper, the viscosity solution…

In this paper we consider a nonlinear system of PDEs coupling the viscous Cahn-Hilliard-Oono equation with dynamic boundary conditions enjoying a similar structure on the boundary. After proving well-posedness of the corresponding initial…

In this paper we study the existence of sufficiently regular representations of Hamilton-Jacobi equations in optimal control theory with the compact control set. We introduce a new method to construct representations for a wide class of…

Autonomous systems have witnessed a rapid increase in their capabilities, but it remains a challenge for them to perform tasks both effectively and safely. The fact that performance and safety can sometimes be competing objectives renders…

We show that viscosity solutions of evolutionary weakly coupled systems of Hamilton--Jacobi equations can be approximated by iterated twisted Lax--Oleinik like operators. We establish convergence to the solution of the iterated scheme and…

We consider an optimal control on networks in the spirit of the works of Achdou et al. (2013) and Imbert et al. (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible…

For a general nonlinear control system, we study the problem of small time local attainability of a target which is the closure of an open set. When the target is smooth and locally the sublevel set of a smooth function, we develop second…

The paper deals with a Bolza optimal control problem for a dynamical system which motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional in this…

We study the well-posedness of Hamilton-Jacobi-Bellman equations on subsets of $\mathbb{R}^d$ in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an…

In the paper, we consider a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We…

In this manuscript we consider optimal control problems of stochastic differential equations with delays in the state and in the control. First, we prove an equivalent Markovian reformulation on Hilbert spaces of the state equation. Then,…

This paper develops a comparison theorem for viscosity solutions of a new class of Hamilton-Jacobi-Bellman (HJB) equations, which is used to solve the separated problem governed by the K-S equation in the Wasserstein space. A distinctive…

We study the viscosity solutions to the first eigenvalue equation. We consider $\Omega$ a bounded B-regular domain in $\mathbb{C}^n$ and we prove that the Dirichlet problem $\Lambda_{1}(D_{\mathbb{C}}^2 u)=f$ in $\Omega$ and $u=\varphi$ on…

The paper studies a system of first order Hamilton-Jacobi equations with discontinuous coefficients, arising from a model of deterministic optimal debt management in infinite time horizon, with exponential discount and currency devaluation.…