Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.     

Optimal control of the viscous Camassa–Holm equation

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.     

Optimal control problem for viscous Cahn-Hilliard equation

This paper is concerned with the viscous Cahn-Hilliard equation, which arises in the dynamics of viscous first order phase transitions in cooling binary solutions. The optimal control under boundary condition is given and the existence of optimal solution to the equation is proved.     

Limit behavior of the global solutions to the Degasperis-Procesi-type equation

In this paper, we study the Degasperis-Procesi equation with a physically perturbation term—a linear dispersion. Based on the global existence result, we show that the solution of the Degasperis-Procesi equation with linear dispersion tends to the solution of the corresponding Degasperis-Procesi equation as the dispersive parameter goes to zero. Moreover, we prove that smooth solutions of the equation have finite propagation speed: they will have compact support if its initial data has this property.     

Some properties of solutions to the weakly dissipative Degasperis-Procesi equation

In this paper, we consider the weakly dissipative Degasperis-Procesi equation. The present paper is concerned with some aspects of existence of global solutions, persistence properties and propagation speed. First we try to discuss the local well-posedness and blow-up scenario, then establish the sufficient conditions on global existence of the solution. Finally, persistence properties on strong solutions and the propagation speed for the weakly dissipative Degasperis-Procesi equation are also investigated.     

Maximum principle for optimal distributed control of viscous weakly dispersive Degasperis–Procesi equation

This paper is concerned with the optimal distributed control of the viscous weakly dispersive Degasperis–Procesi equation in nonlinear shallow water dynamics. It is well known that the Pontryagin maximum principle, which unifies calculus of variations and control theory of ordinary differential equations, sets up the theoretical basis of the modern optimal control theory along with the Bellman dynamic programming principle. In this paper, we commit ourselves to infinite dimensional generalizations of the maximum principle and aim at the optimal control theory of partial differential equations. In contrast to the finite dimensional setting, the maximum principle for the infinite dimensional system does not generally hold as a necessary condition for optimal control. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the controlled viscous weakly dispersive Degasperis–Procesi equation. The necessary optimality condition is established for the problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control of the viscous Dullin–Gottwalld–Holm equation

This paper studies the problem for optimal control of the viscous DGH equation. The existence and uniqueness of weak solution to the equation are proved in a short interval. The optimal control of the viscous DGH equation under boundary condition is given and the existence of optimal solution to the equation is proved.     

Time-Periodic Fitzhugh-Nagumo Equation and the Optimal Control Problems

In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function.     

Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation

We establish the local well-posedness for the viscous Degasperis-Procesi equation. We show that the blow-up phenomena occurs in finite time. Moreover, applying the energy identity, we obtain a global existence result in the energy space.     

Feedback control for nonlinear evolutionary equations with applications

In the paper we are concerned with the feedback control system governed by nonlinear evolutionary equations involving weakly continuous operators. By using the Rothe method and a surjectivity result for weakly continuous operators, we first present the solvability for the evolutionary equation. Then we show the existence of solutions to the feedback control system. We also consider an existence result for an optimal control problem. Moreover, we apply the main results to a class of differential variational inequalities, evolutionary hemivariational inequalities and the non-stationary Navier–Stokes–Voigt equation with a subgradient inclusion condition.     

有限时区上的最优转换和停止问题

讨论了有限时区上的最优转换和停止问题,它是一类同时具备脉冲控制和最优停止特征的最优控制问题.问题的最优值以及最优转换和停止决策可以由具有混合障碍的多维反射倒向随机微分方程的解来刻画.接着考虑了形式更一般的反射倒向随机微分方程并证明了方程解的存在唯一性.     

Optimal impulsive control of piecewise deterministic Markov processes

In this paper, we study the infinite-horizon expected discounted continuous-time optimal control problem for Piecewise Deterministic Markov Processes with both impulsive and gradual (also called continuous) controls. The set of admissible control strategies is supposed to be formed by policies possibly randomized and depending on the past-history of the process. We assume that the gradual control acts on the jump intensity and on the transition measure, but not on the flow. The so-called Hamilton–Jacobi–Bellman (HJB) equation associated to this optimization problem is analyzed. We provide sufficient conditions for the existence of a solution to the HJB equation and show that the solution is in fact unique and coincides with the value function of the control problem. Moreover, the existence of an optimal control strategy is proven having the property to be stationary and non-randomized.     

Degasperis-Procesi方程的孤立尖波解

利用动力系统的定性分析理论对D egasperis-P rocesi方程的孤立尖波解进行了研究.给出了D e-gasperis-P rocesi方程对应行波系统的相图分支,利用相图获得了孤立尖波解和周期尖波解的解析表达式,通过数值模拟给出了部分解的图像.     

有限时区上的最优转换和停止问题

讨论了有限时区上的最优转换和停止问题,它是一类同时具备脉冲控制和最优停止特征的最优控制问题.问题的最优值以及最优转换和停止决策可以由具有混合障碍的多维反射倒向随机微分方程的解来刻画.接着考虑了形式更一般的反射倒向随机微分方程并证明了方程解的存在唯一性.     

On Singular Sets of Local Solutions to p-Laplace Equations

The author proves that the right-hand term of a p-Laplace equation is zero on the singular set of a local solution to the equation. Such a result is used to study the existence of an optimal control problem.     

Orbital stability of peakons for the Degasperis-Procesi equation with strong dispersion

In this paper, we study the orbital stability of the peakons for the Degasperis-Procesi equation with a strong dispersive term on the line. Using the method in [Z. Lin, Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math. 62 (2009) 125-146], we prove that the shapes of these peakons are stable under small perturbations. Some previous results are extended.     

An optimal control problem for two-dimensional Schrödinger equation

In this paper we consider an optimal control problem controlled by three functions which are in the coefficients of a two-dimensional Schrödinger equation. After proving the existence and uniqueness of the optimal solution, we get the Frechet differentiability of the cost functional using Hamilton-Pontryagin function. Then we state a necessary condition to an optimal solution in the variational inequality form using the gradient.     

Optimal control for a sixth order nonlinear parabolic equation

In this paper, we consider the optimal control problem for a sixth order nonlinear parabolic equation, which arising in oil‐water‐surfactant mixtures. Based on Lions' theory, we prove the existence of optimal solution. The optimality system is also established. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient computational method for the optimal control problem for the Burgers equation

Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge–Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.     

Robin-type boundary control problems for the nonlinear Boussinesq type equations

In this paper, we study a linear and a nonlinear boundary control problems arising from viscous flows. The equations are of nonlinear Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the temperature and the salinity. The essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations: time-dependent, coupled and nonlinear. The existence and the conditions of the uniqueness of the solution, for the variational problem, are studied. The control is of linear or nonlinear Robin-type and acts on a part of the boundary during a time T. The cost function measures the distance between the observed and the computed vorticity. The existence of an optimal control in the admissible set of states and controls is proved. A first order necessary conditions of optimality are obtained.