Sufficient Conditions For Optimal Control Research Articles

Necessary and sufficient conditions for optimal controls in viscous flow problems

A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton–Jacobi–Bellman equation characterising the feedback problem. The maximum principle is established by two quite different methods.

Necessary and sufficient conditions for optimal control of quasilinear partial differential systems

First-order necessary and sufficient conditions are obtained for the following quasilinear distributed-parameter optimal control problem: $$max\left\{ {J(u) = \int_\Omega {F(x,u,t) d\omega + } \int_{\partial \Omega } {G(x,t) \cdot d\sigma } } \right\},$$ subject to the partial differential equation $$A(t)x = f(x,u,t),$$ wheret,u,G are vectors andx,F are scalars. Use is made of then-dimensional Green's theorem and the adjoint problem of the equation. The second integral in the objective function is a generalized surface integral. Use of then-dimensional Green's theorem allows simple generalization of single-parameter methods. Sufficiency is proved under a concavity assumption for the maximized Hamiltonian $$H^\circ (x,\lambda ,t) = \max \{ H(x,u,\lambda ,t):u\varepsilon K\} $$ .

First and second order sufficient conditions for optimal control and the calculus of variations

In this paper we develop first and second order sufficient conditions for optimal control and the calculus of variations problems. Our conditions are derived from the Hamilton-Jacobi approach [15, Thm. 2], which was obtained for the generalized problem of Bolza. We do not require any convexity on the data [7] and [11], or that the control setU is polyhedral [14], or that the control function is in the interior ofU [8]. Instead, we assume a certain inequality which is satisfied in each of the above mentioned cases.

Generalized quadratic optimal controllers for linear hereditary systems

A generalization of the quadratic optimal control theory for linear hereditary control systems is presented. The generalization involves admitting delay terms in the cost index so that for example, a quadratic function of the system output can be minimized. Questions of existence of optimal control as well as necessary and sufficient conditions for optimal control are treated. The results have been achieved using a geometric approach based on properties of the set of reachable points.

Sufficient conditions for optimal controls—use of generalized convexity†

Several sufficient conditions for optimality are presented. These conditions are made of necessary conditions sufficiently strengthened with convexity and generalized convexity assumptions. The use of generalized convexity concepts allows one to obtain conditions weaker than some sufficient conditions previously obtained with convexity.

Necessary and Sufficient Conditions for Optimal Control of Semi-Markov Jump Processes

A class of controlled semi-Markov jump processes is defined in this paper. Conditions are found which guarantee that satisfaction of the dynamic programming equations for stochastic control is necessary and sufficient for the minimization of the expected discounted cost of a controlled semi-Markov process over a random time. Terminal costs are included, and the controls are allowed to depend on both the present state of the process and the length of time the process has been in that state.

Sufficient conditions for optimal control with state and control constraints

A monotonicity result is utilized to derive sufficient optimality conditions of considerable generality for an individual trajectory in control theory. The sufficiency theorem embodying these conditions generalizes those of Boltyanskii and Leitmann and is applied to a simple control system to which their sufficiency theorems are not applicable. Conditions on the state equations and state space are completely relaxed. The set of admissible controls is extended to the set of measurable controls and the integrand of the performance index has its membership extended to the class of bounded Borel-measurable functions. The decomposition of the state space is required to be onlyplain denumerable.

A constrained optimal control of nonlinear systems

A new method for optimal control of nonlinear systems with input constraint is discussed in this paper. The system is optimized by minimizing a quadratic performance index. Considering this problem as a nonlinear programming problem, the necessary and sufficient conditions for optimal control are derived, which are later simplified to an integral equation. This integral equation becomes a necessary condition. The existence of a solution of this integral equation is studied, and a method of solving it is discussed. A numerical example is worked out at the end.

Sufficient condition for optimal control of linear systems with nonlinear cost functions

Sufficient condition for optimal control of linear systems with nonlinear cost functions