Optimal control of linear systems represented by integral equations: An iterative method

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 consider a problem of optimal control of the stochastic evolution equation $d\xi = (A(t)\xi + B(t)u)dt + \sigma (t)dw$, on a separable Hilbert space, where $\{ A(t),B(t),\sigma (t),t \geqq 0\} $ are progressively measurable operator-valued random processes with A generally unbounded. We prove the existence and uniqueness of (weak) solutions of the evolution equation. Then we present the existence of optimal controls and necessary conditions of optimality for a quadratic (random) cost function. For optimal feedback controls we solve a random operator Riccati equation and a backward stochastic evolution equation. The backward equation is solved by transposing a random isomorphism generated from a forward evolution equation. The optimal feedback control is given by a random affine transformation of the state. Some examples are presented to indicate usefulness of the results. This work is a partial extension of the results of Bismut [SIAM J. Control Optim., 14 (1976), pp. 419–444; 15 (1977), pp. 1–4] and Bensoussan and Voit [SIAM J. Control Optim., 13 (1975), pp. 904–926].

The Numerical Solution of Integral Equations on the Half-Line

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.

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.

The basic integro-differential equation is subjected to a one-sided finite Laplace transform to obtain linear integral equations of angular distribution of bounding faces. These linear integral equations have been transformed into linear singular integral equations which have been solved exactly to get the emergent distributions from the bounding faces by the theory of linear singular operators. Some solutions of linear singular integral equations have also been derived for future use in radiative transfer problems.

Optimal feedback stabilization policy with asymmetric loss functions

Necessary conditions for optimality in problems with nonstandard cost functionals.- Linear control problems and an existence theorem.- Optimal disturbance rejection and performance robustness in linear systems.- Necessary conditions for optimal disturbance rejection in linear systems.- Synthesis of finite-interval H ? controllers by state space methods.- Worst-case performance measures for linear control problems.- Model reduction with a finite-interval H ? criterion.

In this study, optimal control of a senior coefficient in a heat conduction problem is discussed. Since such problems are severely ill-posed, it is necessary to regularize them. So, we present the regularization process which has a great importance for a optimal control problem in this study. After proving the existence and uniqueness of optimal solution, a necessary condition for optimality is stated. Then, an iterative algorithm is purposed for numerical approximations. Finally, optimal control has been approximately calculated using the obtained results. The outcomes have also been tested with numerical examples.

In this paper, first, we propose an iterative method based on quadrature formula for solving two-dimensional linear fuzzy Fredholm integral equations (2DLFFIE). Then, we prove the error estimation of the method. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, supporting examples are also provided.

This paper concerns optimal control of systems governed by stochastic partial differential equations in which drift and diffusion terms are second- and first-order differential operators, respectively. Necessary conditions for an optimal control are derived for both nondegenerate and degenerate systems, and all the coefficients appearing in the equations are allowed to depend on the control variables. Furthermore, the results obtained in the paper can also be used to derive necessary conditions of optimality for partially observed diffusions with correlation between the signals and the observation noises.

We consider a finite-horizon linear-quadratic optimal control problem where only a limited number of control messages are allowed for sending from the controller to the actuator. To restrict the number of control actions computed and transmitted by the controller, we employ a threshold-based event-triggering mechanism that decides whether or not a control message needs to be calculated and delivered. Due to the nature of threshold-based event-triggering algorithms, finding the optimal control sequence requires minimizing a quadratic cost function over a nonconvex domain. In this paper, we first provide an exact solution to this nonconvex problem by solving an exponential number of quadratic programs. To reduce computational complexity, we then propose two efficient heuristic algorithms based on greedy search and the alternating direction method of multipliers technique. Later, we consider a receding horizon control strategy for linear systems controlled by event-triggered controllers, and we further provide a complete stability analysis of receding horizon control that uses finite-horizon optimization in the proposed class. Numerical examples testify to the viability of the presented design technique.

We consider optimal boundary control of a distributed-parameter system. The system state is described by two parabolic equations of second order, where the coefficients of one equation depend on the gradient of the solution of the second equation. An existence and uniqueness theorem is proved for the optimal control in this problem and the necessary conditions of optimality are derived.

In this paper, the optimal control problem of system described by stochastic McShane differential equations is considered. It is shown that this problem can be reduced to an equivalent optimal control problem of distributed parameter systems of parabolic type with controls appearing in the coefficients of the differential operator. Further, to this reduced problem, necessary conditions for optimality and an existence theorem for optimal controls are given.

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.