Necessary conditions for optimal controls for systems governed by parabolic partial delay-differential equations in divergence form with first boundary conditions

In this paper, we consider the question of necessary conditions for optimality for systems governed by second-order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain controls and delays in their arguments. The second-order parabolic partial delay-differential equation is in divergence form. In Theorem 4.1, we present results on the existence and uniqueness of weak solutions in the sense of Ladyzhenskaya-Solonnikov-Ural'ceva for this class of systems. An integral maximum principle and its point-wise version for the corresponding controlled system are established in Theorem 5.1 and Corollary 5.1, respectively.The authors wish to thank Dr. E. Noussair for his stimulating discussion and valuable comments in the preparation of this paper. Further, they also wish to acknowledge the referee of the paper for his valuable suggestions and comments. The discussion presented in Section 6 is in response to his suggestions.     

On the optimal controls of a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions

Summary In this paper we consider the question of existence of optimal controls for a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions and with controls appearing in the coefficients. In Theorems2.2 and2.3 we present existence and uniqueness of solutions of the first boundary problems. In Theorems3.1 and3.2 we prove that whenever the coefficients of the system converge in the w*-topology (L¹ topology on L^∞) the corresponding solutions converge weakly in an appropriate Sobolev space. Using these basic results we present two theorems (Theorems4.1 and4.2) on the existence of optimal controls. Entrata in Redazione il 21 gennaio 1978.     

Necessary conditions for optimality of singular controls in systems governed by partial differential equations

In this paper, we present a method to obtain necessary conditions for optimality of singular controls in systems governed by partial differential equations (distributed-parameter systems). The method is based on the one developed earlier by the author for singular control problems described by ordinary differential equations. As applications, we consider conditions for optimality of singular controls in a Darboux-Goursat system and in control systems that describe chemical processes.This research was supported in part by the National Science Foundation under Grant No. NSF-MCS-80-02337 at the University of Michigan.The author wishes to express his deep gratitude to Professor L. Cesari for his valuable guidance and constant encouragement during the preparation of this paper.     

On the system governed by parabolic partial delay-differential equations with first boundary conditions

Summary In this paper, we consider a system governed by second order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain delays in their arguments. The second order parabolic partial delay-differential equation is in ? divergence form ?. In Theorem4.1, we present results on the existence and uniqueness of weak solution in the sense of Aronson for this class of systems. In Theorem4.2, we prove that whenever the coefficients and forcing terms converge in the almost everywhere topology the corresponding solutions converge weakly in an appropriate Sobolev space. Entrata in Redazione il 9 novembre 1977.     

On the existence of optimal controls for quasilinear parabolic partial differential equations

A class of systems governed by quasilinear parabolic partial differential equations with first boundary conditions is considered. Existence of solutions for this class of systems and theira priori estimates are established. Further, a theorem on the existence of optimal controls for the corresponding control problem is obtained. Its proof is based on Filippov's implicit functions lemma. The control restraint setU is taken as a measurable multifunction.The authors wish to thank Professor L. Cesari for his most valuable comments and suggestions. In fact, a condition assumed in the original version of this paper was substantially relaxed by him. For details, see Remark 4.1.     

On the optimal control of systems governed by quasilinear integro-partial differential equations of parabolic type

Recently, M. N. O?uztöreli presented certain results on the existence and uniqueness of solutions of systems governed by a linear integro-partial differential equation of parabolic type with delayed arguments. Since his results admit only smooth coefficients, they could not be used directly in the study of the optimal control problems with bounded measurable control variables appearing in the coefficients of the system equations. In this paper, we consider a class of systems described by second-order quasilinear parabolic integro-partial differential equations with all but the second-order coefficients assumed bounded measurable. Our principal results are: Theorem 3.5, which establishes the existence and uniqueness of solutions of this class of systems (with controls in the coefficients), and Theorem 4.4, which gives a necessary condition for optimality for the corresponding controlled system.     

Optimal control of systems governed by time-delayed,second-order,linear, parabolic partial differential equations with a first boundary condition

In this paper, we consider a class of systems governed by time-delayed, second-order, linear, parabolic partial differential equations with first boundary conditions. The existence and uniqueness of solutions of this class of systems are established in Theorem 3.2. A necessary condition for optimality for the corresponding controlled system is presented in Theorem 5.1. For the proof of this theorem, we develop several preparatory results in Sections 2, 3, and 4.     

Optimality conditions for stochastic boundary control problems governed by semilinear parabolic equations

We study the boundary control problems for stochastic parabolic equations with Neumann boundary conditions. Imposing super-parabolic conditions, we establish the existence and uniqueness of the solution of state and adjoint equations with non-homogeneous boundary conditions by the Galerkin approximations method. We also find that, in this case, the adjoint equation (BSPDE) has two boundary conditions (one is non-homogeneous, the other is homogeneous). By these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method.     

Existence of optimal controls for a class of systems governed by differential inclusions on a Banach space

Using Cesari's approach, we prove the existence of optimal controls for a class of systems governed by differential inclusions on a Banach space having the Radon-Nikodym property. Theorem 3.1 gives the existence result for optimal relaxed controls under fairly general assumptions on the system and the admissible controls. This result depends on a fundamental result (Theorem 2.1) that proves the existence of mild solutions of differential inclusions on a Banach space, which has also independent interest. Further, the preparatory results, such as Lemma 3.1 and Lemma 3.2, are also useful in the study of time-optimal and terminal control problems.For illustration of the results, we present two examples, one on distributed controls for a class of systems governed by nonlinear parabolic equations and the other on boundary controls with discontinuous boundary operator.This work is supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.     

Existence of an optimal control for stochastic systems governed by ito equations

We give existence theorems for stochastic control problems with a lower semicontinuous cost functional and governed by Ito equations. We prove that two formulations of the fundamental problem are equivalent, one involving nonanticipative controls and the other involving (measurable) feedback controls. We then use the concept ofconvergence in distribution to prove existence for the first problem, and hence for the second as well. While our work has certain similarities with a paper of Kushner, our techniques are different and lead to more general results.     

Fractional partial differential equations with boundary conditions

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\mathrm{\Omega })$ and $L_1(\mathrm{\Omega })$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.     

Existence theorems without convexity assumptions for optimal control problems governed by parabolic and elliptic systems

We give existence theorems for optimal control problems governed by partial differential equations. We make no convexity assumption with respect to the control variables. The results follow from a combination of the Lyapounov convexity theorem and a condition of concavity with respect to the state variable.     

Existence of solutions for fractional differential equations with multi-point boundary conditions

We discuss the existence of solutions for a nonlinear multi-point boundary value problem of integro-differential equations of fractional order q ∈ (1, 2]. Our analysis relies on the contraction mapping principle and the Krasnoselskii’s fixed point theorem. Example is provided to illustrate the theory.     

一类含非线性边界条件的高阶微分方程解的存在性(英文)

将上下解方法和Leray-Shauder度应用到一类含有非线性边界条件的n阶微分方程,得到了至少存在一个解的结果,并且改进和推广了文献中的某些结果.     

Lower bounds on the cost functional for systems governed by partial differential equations

Given two systems, one governed by a partial differential equation of a parabolic type and the other governed by a partial differential equation of a hyperbolic type, a cost functional is defined. For each system, an inequality is derived. From these inequalities, lower bounds on the cost functional can be estimated.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Quasilinear parabolic stochastic partial differential equations: Existence,uniqueness

We present a direct approach to existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone. The proof of uniqueness is very elementary, based on a new method of applying Itô’s formula for the $L^1$-norm. The proof of existence relies on a recent regularity result and is direct in the sense that it does not rely on the stochastic compactness method.