Leitmann’s direct method for fractional optimization problems

Based on a method introduced by Leitmann [G. Leitmann, A note on absolute extrema of certain integrals, Int. J. Non-Linear Mech. 2 (1967) 55-59], we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control.     

Existence of finitely optimal solutions for infinite-horizon optimal control problems

In this paper, we investigate the existence of finitely optimal solutions for the Lagrange problem of optimal control defined on [0, ) under weaker convexity and seminormality hypotheses than those of previous authors. The notion of finite optimality has been introduced into the literature as the weakest of a hierarchy of types of optimality that have been defined to permit the study of Lagrange problems, arising in mathematical economics, whose cost functions either diverge or are not bounded below. Our method of proof requires us to analyze the continuous dependence of finite-interval Lagrange problems with respect to a prescribed terminal condition. Once this is done, we show that a finitely optimal solution can be obtained as the limit of a sequence of solutions to a sequence of corresponding finite-horizon optimal control problems. Our results utilize the convexity and seminormality hypotheses which are now classical in the existence theory of optimal control.This research forms part of the author's doctoral dissertation written at the University of Delaware, Newark, Delaware under the supervision of Professor Thomas S. Angell.     

Additional necessary conditions for optimal control with state-variable inequality constraints

It is shown that, when the set of necessary conditions for an optimal control problem with state-variable inequality constraints given by Bryson, Denham, and Dreyfus is appropriately augmented, it is equivalent to the (different) set of conditions given by Jacobson, Lele, and Speyer. Relationships among the various multipliers are given.This work was done at NASA Ames Research Center, Moffett Field, California, under a National Research Council Associateship.     

On the existence of sporadically catching-up optimal solutions for infinite-horizon optimal control problems

In this paper, we extend the existence theory of Brock and Haurie concerning the existence of sporadically catching-up optimal solutions for autonomous, infinite-horizon optimal control problems. This notion of optimality is one of a hierarchy of types of optimality that have appeared in the literature to deal with optimal control problems whose cost functionals, described by an improper integral, either diverge or are unbounded below. Our results rely on the now classical convexity and seminormality hypotheses due to Cesari and are weaker than those assumed in the work of Brock and Haurie. An example is presented where our results are applicable, but those of the above-mentioned authors do not.This research forms part of the author's doctoral dissertation, written at the University of Delaware, Newark, Delaware, under the supervision of Professor T. S. Angell.     

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential 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.     

Optimality condition and algorithm with deviation integral for global optimization

To study the integral global minimization, a general form of deviation integral is introduced and its properties are examined in this work. In terms of the deviation integral, optimality condition and algorithms are given. Algorithms are implemented by a properly designed Monte Carlo simulation. Numerical tests are given to show the effectiveness of the method.     

On Global Optimality Conditions for Nonlinear Optimal Control Problems

Let a trajectory and control pair maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g( (T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is shown that this necessary condition for global optimality of turns out to be a sufficient one under the additional assumption of nondegeneracy of the maximum principle for every pair (x,u) from the above-mentioned level set. In particular, if the pair satisfies the Pontryagin maximum principle which is nondegenerate in the sense that for the Hamiltonian H, we have along the pair on [0,T], and if there is no another pair (x,u) such that g(x(T))=g( (T)), then is a global maximizer.     

An optimal control model and algorithm for the deviated well’s trajectory planning

This paper presents a nonlinear piecewise smooth dynamical system of the trajectory of deviated wells according to engineering background. An optimal control model is established and the necessary conditions for optimality are proved via maximum principle. The optimal control problem is solved by a revised Hooke–Jeeves algorithm. The uniform design technique has been incorporated into the revised Hooke–Jeeves algorithm to handle the multimodal function. Computer simulation is used for this paper, and the numerical example illustrates the validity and efficiency of the algorithm. The procedure demonstrates its advantages in practical applications in Liaohe Oil Field.     

Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint

In this paper, we shall investigate the superconvergence property of quadratic elliptical optimal control problems by triangular mixed finite element methods. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite elements and the control is discretized by piecewise constant functions. We prove the superconvergence error estimate of h² in L²-norm between the approximated solution and the interpolation of the exact control variable. Moreover, by postprocessing technique, we find that the projection of the discrete adjoint state is superclose (in order h²) to the exact control variable.     

An SQP method for optimal control of weakly singular Hammerstein integral equations

We investigate local convergence of an SQP method for nonlinear optimal control of weakly singular Hammerstein integral equations. Sufficient conditions for local quadratic convergence of the method are discussed.     

A dynamical model of echinococcosis with optimal control and cost-effectiveness

A dynamical model of echinococcosis transmission with optimal control strategies is first presented. The basic reproduction number of the model is determined and employed to study the global stability of the disease-free and endemic equilibrium points. The optimal control problem is formulated and solved analytically. Numerical simulations show that optimal control strategies could effectively reduce the transmission of echinococcosis. The cost-effectiveness analysis suggests that a combination of health education, anthelmintic treatment, and home slaughter inspection could provide the best cost-effective strategy to control the transmission of echinococcosis. Furthermore, it finds that anthelmintic treatment and environmental disinfection may shorten the time of eliminating the disease. The results may be helpful for prevention and control of echinococcosis in Ganzi Tibetan Autonomous Prefecture, China and other areas of echinococcosis.     

Regularity of flows and optimal control of shear-thinning fluids

We study optimal control problems of systems describing the flow of incompressible shear-thinning fluids. Taking advantage of regularity properties of the flows, we derive necessary optimality conditions under a restriction on the optimal control.     

Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.     

Second-order optimality principle for singular optimal control problems

A new necessary condition for singular optimal control problems is presented in this paper. The condition is simpler to apply than existing conditions and is easily derived from a Taylor series expansion of the performance index.     

Criteria of optimality in the infinite-time optimal control problem

The present paper gives a systematic presentation of different definitions of optimality in the infinite-time optimal control problem. Some of these definitions are new, while others have been used throughout the literature, sometimes with different names. The logical implications between them are clearly stated, corresponding comparison criteria for solutions are defined, and other relations as well as two types of equivalence relations are established.This work was supported in part by Simmons College Fund for Research, Grant No. 201.     

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.     

A filled function method for optimal discrete-valued control problems

In this paper, we present a new approach to solve a class of optimal discrete-valued control problems. This type of problem is first transformed into an equivalent two-level optimization problem involving a combination of a discrete optimization problem and a standard optimal control problem. The standard optimal control problem can be solved by existing optimal control software packages such as MISER 3.2. For the discrete optimization problem, a discrete filled function method is developed to solve it. A numerical example is solved to illustrate the efficiency of our method.     

A virtual control concept for state constrained optimal control problems

A linear elliptic control problem with pointwise state constraints is considered. These constraints are given in the domain. In contrast to this, the control acts only at the boundary. We propose a general concept using virtual control in this paper. The virtual control is introduced in objective, state equation, and constraints. Moreover, additional control constraints for the virtual control are investigated. An error estimate for the regularization error is derived as main result of the paper. The theory is illustrated by numerical tests.