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Register a new userA review from the PDE viewpoint of Hamilton-Jacobi-Bellman Equations Arising in Optimal Control with Vectorial Cost
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Authors
Nikos Katzourakis
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This paper is a review of results on Optimisation which are perhaps not so standard in the PDE realm. To this end, we consider the problem of deriving the PDEs associated to the optimal control of a system of either ODEs or SDEs with respect to a vector-valued cost functional. Optimisation is considered with respect to a partial ordering generated by a given cone. Since in the vector case minima may not exist, we define vectorial value functions as (Pareto) minimals of the ordering. Our main objective is the derivation of the model PDEs which turn out to be parametric families of HJB single equations instead of systems of PDEs. However, this allows the use of the theory of Viscosity Solutions.
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We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ partial_t h(t,x)= uDelta h(t,x)+lambda V(| abla h(t,x)|) +sqrt{D}, eta(t,x), qquad xin{mathbb{R}}^d $$ in $dge 1$ dimensions. The forcing term $eta$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)ge 0$, one finds $V(| abla h|)ltimes | abla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article, a continuation of [24], is dedicated to a generalization of the PDE estimates obtained in the previous article to the case of a deposition rate $V$ with polynomial growth of arbitrary order at infinity, for which in general the Cole-Hopf transformation does not allow any more a comparison to the heat equation. The main tool here instead is the representation of $h$ as the solution of some minimization problem through the Hamilton-Jacobi-Bellman formalism. This sole representation turns out to be powerful enough to produce local or pointwise estimates in ${cal W}$-spaces of functions with locally bounded averages, as in [24], implying in particular global existence and uniqueness of solutions.
Policy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases.
In this article we study ergodic problems in the whole space $mathbb{R}^N$ for a weakly coupled systems of viscous Hamilton-Jacobi equations with coercive right-hand sides. The Hamiltonians are assumed to have a fairly general structure and the switching rates need not be constant. We prove the existence of a critical value $lambda^*$ such that the ergodic eigenvalue problem has a solution for every $lambdaleqlambda^*$ and no solution for $lambda>lambda^*$. Moreover, the existence and uniqueness of non-negative solutions corresponding to the value $lambda^*$ are also established. We also exhibit the implication of these results to the ergodic optimal control problems of controlled switching diffusions.
Let $(Omega, mu)$ be a probability space endowed with an ergodic action, $tau$ of $( {mathbb R} ^n, +)$. Let $H(x,p; omega)=H_omega(x,p)$ be a smooth Hamiltonian on $T^* {mathbb R} ^n$ parametrized by $omegain Omega$ and such that $ H(a+x,p;tau_aomega)=H(x,p;omega)$. We consider for an initial condition $fin C^0 ( {mathbb R}^n)$, the family of variational solutions of the stochastic Hamilton-Jacobi equations $$left{ begin{aligned} frac{partial u^{ varepsilon }}{partial t}(t,x;omega)+Hleft (frac{x}{ varepsilon } , frac{partial u^varepsilon }{partial x}(t,x;omega);omega right )=0 & u^varepsilon (0,x;omega)=f(x)& end{aligned} right .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $omega in Omega$ we have $C^0-lim u^{varepsilon}(t,x;omega)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$left{ begin{aligned} frac{partial v}{partial t}(x)+{overline H}left (frac{partial v }{partial x}(x) right )=0 & v (0,x)=f(x)& end{aligned} right.$$
In this paper, we study the following nonlinear backward stochastic integral partial differential equation with jumps begin{equation*} left{ begin{split} -d V(t,x) =&displaystyleinf_{uin U}bigg{H(t,x,u, DV(t,x),D Phi(t,x), D^2 V(t,x),int_E left(mathcal I V(t,e,x,u)+Psi(t,x+g(t,e,x,u))right)l(t,e) u(de)) &+displaystyleint_{E}big[mathcal I V(t,e,x,u)-displaystyle (g(t, e,x,u), D V(t,x))big] u(d e)+int_{E}big[mathcal I Psi(t,e,x,u)big] u(d e)bigg}dt &-Phi(t,x)dW(t)-displaystyleint_{E} Psi (t, e,x)tildemu(d e,dt), V(T,x)=& h(x), end{split} right. end{equation*} where $tilde mu$ is a Poisson random martingale measure, $W$ is a Brownian motion, and $mathcal I$ is a non-local operator to be specified later. The function $H$ is a given random mapping, which arises from a corresponding non-Markovian optimal control problem. This equation appears as the stochastic Hamilton-Jacobi-Bellman equation, which characterizes the value function of the optimal control problem with a recursive utility cost functional. The solution to the equation is a predictable triplet of random fields $(V,Phi,Psi)$. We show that the value function, under some regularity assumptions, is the solution to the stochastic HJB equation; and a classical solution to this equation is the value function and gives the optimal control. With some additional assumptions on the coefficients, an existence and uniqueness result in the sense of Sobolev space is shown by recasting the backward stochastic partial integral differential equation with jumps as a backward stochastic evolution equation in Hilbert spaces with Poisson jumps.
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