Constrained control of a nonlinear two point boundary value problem,I

In this paper we consider an optimal control problem for a nonlinear second order ordinary differential equation with integral constraints. A necessary optimality condition in form of the Pontryagin minimum principle is derived. The proof is based on McShane-variations of the optimal control, a thorough study of their behaviour in dependence of some denning parameters, a generalized Green formula for second order ordinary differential equations with measurable coefficients and certain tools of convex analysis.Dedicated to Lothar von Wolfersdorf on the occasion of his 60th birthday     

Necessary optimality conditions for infinite dimensional state constrained control problems

This paper is concerned with first order necessary optimality conditions for state constrained control problems in separable Banach spaces. Assuming inward pointing conditions on the constraint, we give a simple proof of Pontryagin maximum principle, relying on infinite dimensional neighboring feasible trajectories theorems proved in [20]. Further, we provide sufficient conditions guaranteeing normality of the maximum principle. We work in the abstract semigroup setting, but nevertheless we apply our results to several concrete models involving controlled PDEs. Pointwise state constraints (as positivity of the solutions) are allowed.     

Solutions of Two Nonlinear Evolution Equations Using Lie Symmetry and Simplest Equation Methods

In this paper, we study two nonlinear evolution partial differential equations, namely, a modified Camassa–Holm–Degasperis–Procesi equation and the generalized Korteweg–de Vries equation with two power law nonlinearities. For the first time, the Lie symmetry method along with the simplest equation method is used to construct exact solutions for these two equations.     

Maximum principle for optimal distributed control of the viscous Dullin-Gottwald-Holm equation

In this paper, the optimal distributed control of the viscous Dullin-Gottwald-Holm equation is investigated. Adopting the Dubovitskii and Milyutin functional analytical approach, we obtain the Pontryagin maximum principle of the system. The necessary optimality condition is established for an optimal control problem in fixed final horizon case. Finally, an illustrative example is also given.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures

We obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton-Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizons. The reflection is given by the normal cone to a proximal retract set.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

条件平均场随机微分方程的最优控制问题

作者研究了一个条件平均场随机微分方程的最优控制问题.这种方程和某些部分信息下的随机最优控制问题有关,并且可以看做是平均场随机微分方程的推广.作者以庞特里雅金最大值原理的形式给出最优控制满足的必要和充分条件.此外,文中给出一个线性二次最优控制问题来说明理论结果的应用.     

Maximum principle for optimal control of neutral stochastic functional differential systems

This paper is concerned with optimal control of neutral stochastic functional differential equations (NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type (VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well.     

Soliton,kink and antikink solutions of a 2-component of the Degasperis–Procesi equation

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the traveling wave solutions of a 2-component of the Degasperis–Procesi equation. The expressions for smooth soliton, kink and antikink solutions are obtained.     

Some Facts about the Ramsey Model

In modeling the dynamics of capital, the Ramsey equation coupled with the Cobb–Douglas production function is reduced to a linear differential equation by means of the Bernoulli substitution. This equation is used in the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. This result enriches our understanding of the model analysis in the optimal control framework.     

Controllability and Time Optimal Control for Low Reynolds Numbers Swimmers

The aim of this paper is to tackle the self-propelling at low Reynolds number by using tools coming from control theory. More precisely we first address the controllability problem: “Given two arbitrary positions, does it exist “controls” such that the body can swim from one position to another, with null initial and final deformations?”. We consider a spherical object surrounded by a viscous incompressible fluid filling the remaining part of the three dimensional space. The object is undergoing radial and axi-symmetric deformations in order to propel itself in the fluid. Since we assume that the motion takes place at low Reynolds number, the fluid is governed by the Stokes equations. In this case, the governing equations can be reduced to a finite dimensional control system. By combining perturbation arguments and Lie brackets computations, we establish the controllability property. Finally we study the time optimal control problem for a simplified system. We derive the necessary optimality conditions by using the Pontryagin maximum principle. In several particular cases we are able to compute the explicit form of the time optimal control and to investigate the variation of optimal solutions with respect to the number of inputs.     

Stability of peakons for the Degasperis‐Procesi equation

The Degasperis‐Procesi equation can be derived as a member of a one‐parameter family of asymptotic shallow‐water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa‐Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis‐Procesi equation on the line. By constructing a Lyapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations. © 2007 Wiley Periodicals, Inc.     

Stochastic Maximum Principle for Optimal Control of SPDEs

We prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a stochastic partial differential equation driven by a finite dimensional Wiener process. The equation is formulated in a semi-abstract form that allows direct applications to a large class of controlled stochastic parabolic equations. We allow for a diffusion coefficient dependent on the control parameter, and the space of control actions is general, so that in particular we need to introduce two adjoint processes. The second adjoint process takes values in a suitable space of operators on L ⁴.     

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.     

Smooth and Nonsmooth Lipschitz Controls for a Class of Vector Differential Equations

This paper is devoted to the study of a class of control problems associated to a nonlinear second-order vector differential equation with pointwise state constraints. The control is realized via a function of the state. We extend the results of Akkouchi, Bounabat, and Goebel to vector differential equations and furthermore consider the more general case. Under proper conditions, we prove the existence of optimal controls in the class of Lipschitz functions and obtain an optimality condition which looks somehow like the Pontryagin maximum principle for a smooth optimal control function. For a nonsmooth optimal control function, we derive a suboptimality condition by means of the Ekeland variational principle.Communicated by M. J. BalasThis work was supported by 985 Project of Jilin University. The author thanks Professor Yong Li for valuable suggestions. He also thanks Professor M. J. Balas and the anonymous referees for their comments.     

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler–Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass–Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.     

Optimal control of the viscous weakly dispersive Degasperis-Procesi equation

In this paper, we study the optimal control problem for the viscous weakly dispersive Degasperis-Procesi equation. We deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Then, according to optimal control theories and distributed parameter system control theories, the optimal control of the viscous weakly dispersive Degasperis-Procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive Degasperis-Procesi equation is proved.     

Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations

This paper deals with the Pontryagin maximum principle for optimal control problems governed by 3D Navier–Stokes equations with pointwise control constraint. The obtained result is proved by using some results on regularity of solutions of the Navier–Stokes equations and techniques of optimal control theory.