LQ Optimal Control of First-Order Hyperbolic PDE Systems with Final State Constraints

This paper studies the linear-quadratic (LQ) optimal control problem of a class of systems governed by the first-order hyperbolic partial differential equations (PDEs) with final state constraints. The main contribution is to present the solvability condition and the corresponding explicit optimal controller by using the Lagrange multiplier method and the technique of solving forward and backward partial differential equations (FBPDEs). In particular, the result is reduced to the case with zero-valued final state constraints. Several numerical examples are provided to demonstrate the performance of the designed optimal controller

Exact controllability for a refined Stochastic Wave Equation

In this paper, we obtain the exact controllability for a refined stochastic wave equation with three controls by establishing a novel Carleman estimate for a backward hyperbolic-like operator. Compared with the known result, the novelty of this paper is twofold: (1) Our model contains the effects in the drift terms when we put controls directly in the diffusion terms, which is more sensible for practical applications; (2) We provide an explicit description of the waiting time which is sharp in the case of dimension one and is independent of the coefficients of lower terms

New global Carleman estimates and null controllability for forward/backward semi-linear parabolic SPDEs

In this paper, we study the null controllability for some linear and semi-linear parabolic SPDEs involving both the state and the gradient of the state. To start with, an improved global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with general random coefficients and $L^2$-valued source terms is derived. Based on this, we further develop a new global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with $H^{-1}$-valued source terms, which enables us to deal with the global null controllability for linear backward (resp. forward) parabolic SPDEs with gradient terms. As byproduct, a special energy-type estimate for the controlled system that explicitly depends on the parameters $\lambda,\mu$ and the weighted function $\theta$ is obtained. Furthermore, by employing a fixed-point argument, we extend the previous linear controllability results to some semi-linear backward (resp. forward) parabolic SPDEs

On Existence of L1L^1-solutions for Coupled Boltzmann Transport Equation and Radiation Therapy Treatment Optimization

The paper considers a linear system of Boltzmann transport equations modelling the evolution of three species of particles, photons, electrons and positrons. The system is coupled because of the collision term (an integral operator). The model is intended especially for dose calculation (forward problem) in radiation therapy. It, however, does not apply to all relevant interactions in its present form. We show under physically relevant assumptions that the system has a unique solution in appropriate ($L^1$-based) spaces and that the solution is non-negative when the data (internal source and inflow boundary source) is non-negative. In order to be self-contained as much as is practically possible, many (basic) results and proofs have been reproduced in the paper. Existence, uniqueness and non-negativity of solutions for the related time-dependent coupled system are also proven. Moreover, we deal with inverse radiation treatment planning problem (inverse problem) as an optimal control problem both for external and internal therapy (in general $L^p$-spaces). Especially, in the case $p=2$ variational equations for an optimal control related to an appropriate differentiable convex object function are verified. Its solution can be used as an initial point for an actual (global) optimization.Comment: Corrected typos. Added a new section 3. Revised the argument of Example 7.