Approximation methods in optimal control problems for nonlinear infinite-dimensional systems

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

In this paper, our primary focus is on studying the properties of self-similar solutions to the Cauchy problem. We specifically examine the behavior of these solutions in a double nonlinear time-dependent parabolic equation and their absorption in a non-homogeneous medium. Through the research of the topic matter, our aim is to deliver a more thorough comprehension of the finite speed perturbations propagation in the solution of the Cauchy problem for a nonlinear parabolic equation. This establishment of a property is essential to understand the dynamic nature of these equations. Furthermore, we delve into the self-similar analysis of the solution, which allows us to ascertain the condition of Fujita type global solvability for the Cauchy problem in a double nonlinear degenerate-type parabolic equation within a non-homogeneous medium. This analysis provides valuable insights into the behavior and potential solvability of these equations on a global mean. Additionally, we establish estimates for weak solutions depending on the growing density and the value of numerical parameters. By establishing these estimates, we provide a more comprehensive understanding of the behavior of the solutions in different scenarios.

An optimal control problem for systems described by Fourier problem for nonlinear parabolic equations is studied. Control functions occur in the coefficients of the state equations. The existence of the optimal control in the case of final observation is proved.

This paper discusses feedback control problems for affine nonlinear singular systems. Namely, the problems of regularization (regular singular systems have for a particular initial state and input, a unique solution), non-interaction and exact linearization. First, an algorithm providing necessary and sufficient conditions for the regularizability of the affine nonlinear singular systems is introduced. This algorithm resembles the well-known constrained dynamics algorithm for affine nonlinear systems. For regularizable affine nonlinear singular systems, necessary and sufficient conditions for the solvability of the non-interacting control problem are derived. They are based on another algorithm producing a sequence of integers, the generalization of the well known notion of relative degree for usual affine nonlinear systems. Finally, the problem of feedback exact linearization of the dynamic part of singular systems is briefly addressed. An example is provided to illustrate the main results.

Blow-up problems for nonlinear parabolic equations on locally finite graphs

Abstract. We study the inverse problem for nonlinear parabolic equation with variable thermal conductivity a ( x , t ) ${a(x,t)}$ which is a ill-posed problem. We regularize this problem by using the truncated integral method and then obtain the error estimate between the exact solution and the regularized solution. A numerical experiment is given for illustrating our method.

In this paper, an on-line adaptive dynamic programming (ADP) algorithm is presented to solve the optimal control problem for continuous-time nonlinear systems with control constraints. First, a suitable non-quadratic performance function is introduced to confront control constraints, and then we present an on-line ADP algorithm in which dynamic programming techniques and neural networks are used to solve the optimal control problem for nonlinear systems with control constraints, which is completely different with the traditional ADP algorithm using off-line training, and the corresponding convergence proof is given. Finally, a simulation examples is given to demonstrate the convergence and feasibility of the proposed algorithm.

An efficient method is introduced for solving optimal control and optimal parameter selection problems of nonlinear differential-algebraic systems involving general constraints. These infinite dimensional problems are converted into the finite dimensional optimization problems by using the control parametrization technique. Moreover, the transformed time locations for the control variables are included in such problems. As a result, optimal control problems become acausal optimal parameter selection problems. A modified version of differential evolution is introduced to solve these problems. The integration of the penalty functions is used to ensure the solution onto the feasible domain of the problems. Two industrial systems, a chemical reactor and a robotics arm, are used to illustrate the robustness and effectiveness of the method. The control effects can be improved by selecting the proper time locations as observed from the simulated results.

We study boundary-value problems with Dirichlet conditions for nonlinear parabolic equations with variable delay (i.e., delay is a function of time) and degeneration at the initial time. The existence and uniqueness of the classical solution of this problem are proved. The a priori estimates of this solution are obtained.

This paper focuses on the adaptive reinforcement learning-based optimal control problem for standard nonstrict-feedback nonlinear systems with the actuator fault and an unknown dead zone. To simultaneously reduce the computational complexity and eliminate the local optimal problem, a novel neural network weight updated algorithm is presented to replace the classic gradient descent method. By utilizing the backstepping technique, the actor critic-based reinforcement learning control strategy is developed for high-order nonlinear nonstrict-feedback systems. In addition, two auxiliary parameters are presented to deal with the input dead zone and actuator fault respectively. All signals in the system are proven to be semi-globally uniformly ultimately bounded by Lyapunov theory analysis. At the end of the paper, some simulation results are shown to illustrate the remarkable effect of the proposed approach.

A relation between the output regulation and the observer design for nonlinear systems

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

On the well-posedness of a Fourier problem for nonlinear parabolic equations of higher order with variable exponents of nonlinearity

Coefficient inverse problems are reformulated to a unified integral differential equation. The presented method of conversion of the considered inverse problems to a unified Volterra integral-differential equation gives an opportunity to distribute the acquired results also to analogous inverse problems for non-linear parabolic equations of different types.

On the uniqueness of the solution of a problem for non-linear parabolic equations without initial conditions