This paper proposes a dominated growth rate method which is used to solve a partially observable optimal control problem driven by forward-backward stochastic differential equation. In the problem, state noise is correlated to observation noise, and drift term of the observation equation is not necessarily bounded with respect to state process. This results in state equation and adjoint equation with unbounded stochastic coefficients, whose well-posedness and high-order moment estimates of solutions are obtained. Combining these estimates with a dual technique, a maximum principle for optimal control process of the problem is established. An extended nonlinear optimality problem is explicitly solved, and the corresponding optimal control process is expressed as a linear feedback of finite dimensional optimal filtering of state process under certain circumstance. A government debt stabilization problem is used to demonstrate the theoretical results obtained here.
We consider Lipschitz continuous viscosity solutions to evolutive Hamilton-Jacobi equations. Under a condition of strict convexity of the Hamiltonian, we show that there exists a notion of strong trace of the gradient of the solution. This result is based on a Liouville-type result of classification of global solutions on the half space. Under zero Dirichlet boundary condition, we show that the solution only depends on the normal variable. As a consequence, we show that the existence of a pointwise tangential gradient implies existence of a pointwise normal gradient. For the Liouville-type result, and when the Hamiltonian is not convex, we give a counter-example with a solution which is not one-dimensional. We give two applications. On the one hand, for the classical stationary Dirichlet problem on a bounded domain, we show the existence of a closed subset of the boundary of the domain, where Taylor expansion of the solution is uniform. On the other hand, for Hamilton-Jacobi equations on a network, we show that the space derivative of the solution has a trace at each node, which satisfies a natural germ condition.
This work addresses the controllability of the energy of quantum bounded states through domain deformations in the two-dimensional framework. We approach the controllability question from a practical point of view, with the primary goal of providing simple and implementable control processes involving moving rectangles. We numerically validate the feasibility of our controls and analyze the energy transitions that occur in quantum states confined within specific deformations. This work presents two possible approaches for numerically implementing domain deformations. We show that one can either apply a suitable change of variables to fix the domain, or simulate the deformation using high-intensity confining potentials.
In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in a two-dimensional bounded domain. We formulate a distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier-Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite-dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via the comparison principle.
This paper studies the feedback stabilizability of linear abstract control systems with unbounded control operators, whose state operators generate analytic semigroups. Under an almost exponential decay condition that ensures the existence of bounded feedback operators, we establish several equivalent characterizations of stabilizability. Our main tool is an extended-state-space construction in which the control operator becomes admissible, combined with a Hautus-type resolvent (frequency-domain) inequality. As an application, we recover and extend existing results to settings in which the semigroup generated by the state operator is non-compact and the control operator may fail to be admissible.
