Robust control problems associated with time-varying delay nonlinear parabolic equations

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Starting from the 1st order necessary conditions for optimality, we derive a Hamiltonian system for the state and adjoints with split boundary conditions. Then, recognizing the two point boundary value problem as a canonical transformation, we employ generating functions to find the optimal feedback control as well as the optimal trajectory. Though we formulate the optimal control problem in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a non-linear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two point boundary value problem using a forward shooting method.

We study a model for the exploitation of renewable stocks developed in [5]. In this particular control problem, the control law contains a measurable and an impulsive control component. We formulate Pontryagin's maximum principle for this kind of control problems (see [3]), proving first order necessary conditions of optimality. Manipulating the correspondent Lagrange multipliers we manage to define two special switch functions, that allow the complete description the optimal trajectories and control policies, for all possible initial conditions in the phase plane.

In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.

An earlier developed theory of first and second order necessary conditions for optimality in Banach spaces which gives nondegenerate conditions also in the case of nonregular operators is applied to the optimal control problem. An extended version of the maximum principle as well as a second order optimality condition are presented which have a nontrivial form in the case of abnormal problems. >

This chapter concerns some of the most important optimal control problems related to age-dependent population dynamics. The main goal is to prove the existence of an optimal control and to obtain first order necessary conditions of optimality. These conditions allow calculation or approximation of the optimal control and the optimal value of the cost functional. We shall also present a fractional step scheme for a certain nonlinear problem. This scheme gives a very good method of approximating the optimal control.

We consider a class of Dynkin games in the case where the underlying process evolves according to a one-dimensional but otherwise general diffusion. We establish general conditions under which both the value and the saddle point equilibrium exist and under which the exercise boundaries characterizing the saddle point strategy can be explicitly characterized in terms of a pair of standard first order necessary conditions for optimality. We also analyze those cases where an extremal pair of boundaries exists and investigate the overall impact of increased volatility on the equilibrium stopping strategies and their values.

A recent result giving the structure of the Volterra kernels in terms of Lie-brackets of vector fields allows a new formulation of some higher order necessary conditions of optimality. These conditions are equivalent to the generalized Legendre-Clebsch and Jacobson conditions.

In this study is presented an algorithm for optimal reactive dispatch problem. The algorithm is based on Newton's method and Primal-dual logarithmic barrier method. A Lagrangian function is associated with the modified problem. The first order necessary conditions for optimality are reached by Newton's method, and by updating the penalty and barrier terms. The proposed approach does not have to identify the biding constraints set and can be utilized to infeasible starting point. The effectiveness of the proposed approach has been examined by solving the Brazilian 53-bus and IEEE 162-bus systems.

First order necessary conditions for optimality are the Pontryagin Maximal Principle (PMP) or the Euler-Lagrange and Legendre-Clebsch conditions. In many problems of practical importance, particularly those when the control enters linearly, these conditions are inconclusive for determining the optimal control. In 1964 Kelley [1] discovered a generalization of the Legendre-Clebsch condition and this was extended by Robbins [2,3], Tait [4], Goh [5,6], Kopp and Moyer [7], and Kelley, Kopp and Moyer [8]. An excellent survey of these papers and related results is Gabasov and Kirillova [9].

In this paper, a finite branch-and-bound algorithm is developed for the minimization of a concave power law over a polytope. Linear terms are also included in the objective function. Using the first order necessary conditions of optimality, the optimization problem is transformed into an equivalent problem consisting of a linear objective function, a set of linear constraints, a set of convex constraints, and a set of bilinear complementary constraints. The transformed problem is then solved using a finite branch-and-bound algorithm that solves two convex problems at each of its nodes. The method is illustrated by means of an example from the literature.

Necessary conditions of optimality for the time optimal problem for nonlinear monotone parabolic equations are presented. For some problems governed by a parabolic variational inequality, these conditions are applied to a penalty problem associated with the inequality. Optimality conditions for the time optimal problem for the variational inequality are derived by passing to the limit in the penalizing term.

The present paper is dedicated to the investigation of optimal control problems whose behavior is described by quasilinear differential equations of first order on the plane with nonlocal Bitsadze–Samarski boundary conditions. A theorem of the existence and uniqueness of a generalized solution in the space \(C_{\mu }(\overline{G})\) is proved for quasilinear differential equations; necessary conditions of optimality are obtained in terms of the principle of maximum; the Bitsadze–Samarski boundary value problem is proved for a linear differential equation of first order; the existence of a solution in the space \(C_{\mu }^{p}(\overline{G})\) is proved and an a priori estimate is derived. A theorem on the necessary and sufficient condition of optimality is proved for a linear optimal control problem.

Difference inclusions provide a discrete-time analogue of differential inclusions, which in turn play an important role in the theories of optimal control, implicit differential equations, and invariance and viability, to name a few. In this paper we: (i) introduce a framework suitable for the study of difference inclusions for which the state evolves on a manifold; (ii) use this framework to develop necessary conditions for optimality for a broad class of discrete-time problems of dynamic optimization in which the state evolves on a manifold M. The necessary conditions for optimality we derive include the case for which the state $$q_i$$ is subject to constraints $$q_i \in S_i \subseteq M$$ , for $$S_i$$ a closed set. The resulting necessary conditions for optimality appear as discrete-time versions of the Euler–Lagrange inclusion studied by Ioffe (in Trans Am Math Soc 349(7):2871–2900, 1997), Ioffe and Rockafellar (in Calc Var Partial Differ Equ 4(1):59–87, 1996), Mordukhovich (in SIAM J Control Optim 33(3):882–915, 1995), Mordukhovich (in Variational analysis and generalized differentiation II: applications. Springer, Berlin, 2006), and Vinter and Zheng (in SIAM J Control Optim 35(1):56–77, 1997) generalized in a natural way to the case in which the state is evolving on a manifold.

In this work, we discuss normal forms of necessary conditions of optimality (NCO) for optimal control problems subject to pathwise state constraints and for problems in the calculus of variations with inequality constraints. It is known that standard forms of the NCO may fail to provide information that is useful to identify optimal solutions, namely when the multiplier associated with the objective function takes the value zero. The normal forms of the NCO guarantee that the conditions remain always informative, which is of importance in critical applications where decisions based on optimization are taken, such as autonomous systems. Based on a previous nondegenerate maximum principle for optimal control problems, we extend the strengthness of these conditions to normality while applying them to the particular case of calculus of variations problems. We compare our results with existent normal forms of NCO for dynamic optimization problems and conclude that, when applied to calculus of variations problems, we may say that, under similar conditions, we can apply such result to a wider class of problems, having less regularity on the data.

In this article we apply the optimal and the robust control theory to the sine-Gordon equation. In our case the control is given by the boundary conditions and we work in a finite time horizon. We present at the beginning the optimal control problem and we derive a necessary condition of optimality and we continue by formulating a robust control problem for which existence and uniqueness of solutions are derived.