Hamilton-Jacobi theory over time scales and applications to linear-quadratic problems

In this paper we first derive the verification theorem for nonlinear optimal control problems over time scales. That is, we show that the value function is the only solution of the Hamilton-Jacobi equation, in which the minimum is attained at an optimal feedback controller. Applications to the linear-quadratic regulator problem (LQR problem) gives a feedback optimal controller form in terms of the solution of a generalized time scale Riccati equation, and that every optimal solution of the LQR problem must take that form. A connection of the newly obtained Riccati equation with the traditional one is established. Problems with shift in the state variable are also considered. As an important tool for the latter theory we obtain a new formula for the chain rule on time scales. Finally, the corresponding LQR problem with shift in the state variable is analyzed and the results are related to previous ones.     

The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures

We obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton-Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizons. The reflection is given by the normal cone to a proximal retract set.     

Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems

The aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics. In the first part we characterize the value function for the Mayer problem and the supremum cost problem using viscosity tools and the notion of ε-viability (near viability). These value functions are given with respect to discontinuous cost functionals. In the second part we obtain results describing the ε-viability (near viability) of singularly perturbed control systems.     

Periodic solutions for dynamic equations on time scales

Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented.     

The periodic problem for curvature-like equations with asymmetric perturbations

We discuss existence and multiplicity of solutions of the periodic problem for the curvature-like equation

     

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.     

Constrained control of a nonlinear two point boundary value problem,I

In this paper we consider an optimal control problem for a nonlinear second order ordinary differential equation with integral constraints. A necessary optimality condition in form of the Pontryagin minimum principle is derived. The proof is based on McShane-variations of the optimal control, a thorough study of their behaviour in dependence of some denning parameters, a generalized Green formula for second order ordinary differential equations with measurable coefficients and certain tools of convex analysis.Dedicated to Lothar von Wolfersdorf on the occasion of his 60th birthday     

Robust optimal feedback for terminal linear-quadratic control problems under disturbances

We consider a linear dynamic system in the presence of an unknown but bounded perturbation and study how to control the system in order to get into a prescribed neighborhood of a zero at a given final moment. The quality of a control is estimated by the quadratic functional. We define optimal guaranteed program controls as controls that are allowed to be corrected at one intermediate time moment. We show that an infinite dimensional problem of constructing such controls is equivalent to a special bilevel problem of mathematical programming which can be solved explicitely. An easy implementable algorithm for solving the bilevel optimization problem is derived. Based on this algorithm we propose an algorithm of constructing a guaranteed feedback control with one correction moment. We describe the rules of computing feedback which can be implemented in real time mode. The results of illustrative tests are given.     

An efficient algorithm for maximum entropy extension of block-circulant covariance matrices

This paper deals with maximum entropy completion of partially specified block-circulant matrices. Since positive definite symmetric circulants happen to be covariance matrices of stationary periodic processes, in particular of stationary reciprocal processes, this problem has applications in signal processing, in particular to image modeling. In fact it is strictly related to maximum likelihood estimation of bilateral AR-type representations of acausal signals subject to certain conditional independence constraints. The maximum entropy completion problem for block-circulant matrices has recently been solved by the authors, although leaving open the problem of an efficient computation of the solution. In this paper, we provide an efficient algorithm for computing its solution which compares very favorably with existing algorithms designed for positive definite matrix extension problems. The proposed algorithm benefits from the analysis of the relationship between our problem and the band-extension problem for block-Toeplitz matrices also developed in this paper.     

Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost

We study viscosity solutions of Hamilton-Jacobi equations that arise in optimal control problems with unbounded controls and discontinuous Lagrangian. In our assumptions, the comparison principle will not hold, in general. We prove optimality principles that extend the scope of the results of [23] under very general assumptions, allowing unbounded controls. In particular, our results apply to calculus of variations problems under Tonelli type coercivity conditions. Optimality principles can be applied to obtain necessary and sufficient conditions for uniqueness in boundary value problems, and to characterize minimal and maximal solutions when uniqueness fails. We give examples of applications of our results in this direction.     

Solvability of Sturm-Liouville problems on time scales at resonance

Under the appropriate growth conditions on the inhomogeneous term, we present the existence of the nontrivial solution for Sturm-Liouville boundary value problems on time scales at resonance by making use of coincidence degree theory of Mawhin. The interesting point is the nonlinearity f which is involved with first-order Δ-derivative explicitly.     

Approximate controllability of evolution systems with nonlocal conditions

We study the approximate controllability for the abstract evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the approximate controllability of the semilinear evolution equation. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to show the application of our result.     

Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints

This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control.     

Model of chemotaxis with threshold density and singular diffusion

A quasilinear singular parabolic system corresponding to recent models of chemotaxis in which (1) there is an impassable threshold for the density of cells and (2) the diffusion of cells becomes singular (fast or superdiffusion) when the density approaches the threshold. It is proved that for some range of parameters describing the relation between the diffusive and the chemotactic component of the cell flux there are global-in-time classical solutions which in some cases are separated from the threshold uniformly in time. Global-in-time weak solutions in the case of fast diffusion and the set of stationary states are studied as well. The applications of the general results to particular models are shown.     

The maximum principle in optimal control problems with concentrated and distributed delays in controls

In the present work there has been posed and studied a general nonlinear optimal problem and a quasi-linear optimal problem with fixed time and free right end. It contains absolutely continuous monotone delays in phase coordinates and absolutely continuous monotone and distributed delays in controls. For these problems the necessary and, respectively, sufficient conditions of optimality in the form of the maximum principle have been proved.     

Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales

In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales

     

Partially stabilizing LQ-optimal control for stabilizable semigroup systems

The Linear-Quadratic optimal control problem with a partial stabilization constraint (LQPS) is considered for exponentially stabilizable infinite dimensional semigroup state-space systems with bounded sensing and control (having their transfer function with entries in the algebra . It is reported that the LQPS-optimal state-feedback operator is related to a nonnegative self-adjoint solution of an operator Riccati equation and it can be identified (1) by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper-stable transfer function algebra _, and (2) by obtaining any constant solution of a diophantine equation over _. These theoretical results are applied to a simple model of heat diffusion, leading to an approximation procedure converging exponentially fast to the LQPS-optimal state feedback operator.