准速度在最优控制中的一些应用

一类新的演化多值拟变分不等式 I：存在性与非光滑最优控制

A new class of evolution multivalued quasi-variational inequalities I: existence and nonsmooth optimal control

本文针对一类新的演化多值拟变分不等式，在温和假设下建立了解存在性与解集紧性的新框架，并研究其控制的非线性非光滑最优控制问题，给出了可解性的充分条件，为相关最优控制与参数辨识提供了模型基础。

Plateau拟极小化子到边界的最优正则性

Optimal regularity up to the boundary for Plateau-quasi-minimizers

本文研究带边界条件的Plateau拟极小化子的最优正则性，证明其可由具有Ahlfors正则边界的双John域刻画，并建立了到边界的Ahlfors正则性与一致可整流性。

Sufficient conditions for the value function and optimal strategy to be even and quasi-convex

Sufficient conditions are identified under which the value function and the optimal strategy of a Markov decision process (MDP) are even and quasi-convex in the state. The key idea behind these conditions is the following. First, sufficient conditions for the value function and optimal strategy to be even are identified. Next, it is shown that if the value function and optimal strategy are even, then one can construct a "folded MDP" defined only on the non-negative values of the state space. Then, the standard sufficient conditions for the value function and optimal strategy to be monotone are…

A Quasi Time Optimal Receding Horizon Control

This paper presents a quasi time optimal receding horizon control algorithm. The proposed algorithm generates near time optimal control when the state of the system is far from the target. When the state attains a certain neighbourhood of the aim, it begins the adaptation of the cost function. The purpose of this adaptation is to move from the time optimal control to the stabilizing control. Sufficient conditions for the stability of the closed loop system and the manner of the adaptation of the cost function have been given. Considerations are illustrated with examples. Keywords: Receding hor…

Optimal Control of a Quasi-Variational Sweeping Process

The paper addresses the study of a class of evolutionary quasi-variational inequalities of the parabolic type arising in the formation and growth models of granular and cohensionless materials. Such models and their mathematical descriptions are highly challenging and require powerful tools of their analysis and implementation. We formulate a space-time continuous optimal control problem for a basic model of this type, develop several regularization and approximation procedures, and establish the existence of optimal solutions for the time-continuous and space-discrete problem. Viewing a versi…

拟线性抛物型方程的最优控制问题及其线性近似

Optimal control problems for quasi-linear parabolic equations and their linear approximations

该研究针对热扩散模型，精确了拟线性抛物型方程最优控制问题的线性近似，证明小数据下拟线性与线性问题的最优控制及轨迹差为高阶无穷小，内点情形近似阶最优。

No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from countably many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish "no-gap" second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i. e., for which the sufficient condition only differs from the necessary one in the fact that …

Quasi-optimal robust stabilization of control systems

In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise.

One-dimensional optimal control problems for time-discrete constrained quasilinear diffusion equations of Allen--Cahn types

In this paper, we consider a class of optimal control problems for a one-dimensional time-discrete constrained quasilinear diffusion state-systems of singular Allen--Cahn types and its regularized approximating problems. We note that the control parameter for each system is given by physical temperature. The principal part of this paper is started with the verification of a Key-Theorem dealing with the decompositions of the subdifferentials of the governing convex energies of the state-systems. On this basis, we will prove five Main Theorems, concerned with: the solvability and precise regular…

A Learning-boosted Quasi-Newton Method for AC Optimal Power Flow

Power grid operators typically solve large-scale, nonconvex optimal power flow (OPF) problems throughout the day to determine optimal setpoints for generators while adhering to physical constraints. Despite being at the heart of many OPF solvers, Newton-Raphson can be slow and numerically unstable. To reduce the computational burden associated with calculating the full Jacobian and its inverse, many Quasi-Newton methods attempt to find a solution to the optimality conditions by leveraging an approximate Jacobian matrix. In this paper, a Quasi-Newton method based on machine learning is presente…

Stability of the solution set of quasi-variational inequalities and optimal control

For a class of quasi-variational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to {monotone} perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimiz…

w-Invexity and Optimality Problem

We define w-invex set, w-preinvex, w-strictly preinvex, w-quasi preinvex, w-strictly quasi preinvex, w-semi-strictly quasi preinvex, and w-pre pseudo-invex functions in this context. And these form a class of real functions, which is the generalization of a family of preinvex functions. Here, we provide a thorough analysis of the core characteristics of these functions, along with numerous examples that help to illustrate the idea. Finally, w-quasi preinvex, w-strictly preinvex, and w-strictly quasi preinvex functions are used to analyze the optimization problems.

Optimal control of a non-smooth quasilinear elliptic equation

This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a PC^1 function with countably many points of nondifferentiability, we show that both conditions ar…

The quasi-neutral limit in optimal semiconductor design

We study the quasi-neutral limit in an optimal semiconductor design problem constrained by a nonlinear, nonlocal Poisson equation modelling the drift diffusion equations in thermal equilibrium. While a broad knowledge on the asymptotic links between the different models in the semiconductor model hierarchy exists, there are so far no results on the corresponding optimization problems available. Using a variational approach we end up with a bi-level optimization problem, which is thoroughly analysed. Further, we exploit the concept of Gamma-convergence to perform the quasi-neutral limit for the…