Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ? with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ? in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ? in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ?. Finally, we define partial quadratic functionals associated with ? and a (strong) regularity of ?, which we relate to the positivity and nonnegativity of ?.     

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.     

Equivalent conditions to the nonnegativity of a quadratic functional in discrete optimal control

In this paper we provide a characterization of the nonnegativity of a discrete quadratic functional ? with fixed right endpoint in the optimal control setting. This characterization is closely related to the kernel condition earlier introduced by M. Bohner as a part of a focal points definition for conjoined bases of the associated linear Hamiltonian difference system. When this kernel condition is satisfied only up to a certain critical index m, the traditional conditions, which are the focal points, conjugate intervals, implicit Riccati equation, and partial quadratic functionals, must be replaced by a new condition. This new condition is determined to be the nonnegativity of a block tridiagonal matrix, representing the remainder of ? after the index m, on a suitable subspace. Applications of our result include the discrete Jacobi condition, a unification of the nonnegativity and positivity of ? into one statement, and an improved result for the special case of the discrete calculus of variations. Even when both endpoints of ? are fixed, this paper provides a new result. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Riccati inequality and other results for discrete symplectic systems

In this paper we establish several new results regarding the positivity and nonnegativity of discrete quadratic functionals F associated with discrete symplectic systems. In particular, we derive (i) the Riccati inequality for the positivity of F with separated endpoints, (ii) a characterization of the nonnegativity of F for the case of general (jointly varying) endpoints, and (iii) several perturbation-type inequalities regarding the nonnegativity of F with zero endpoints. Some of these results are new even for the special case of discrete Hamiltonian systems.     

Discrete Optimal Control: Second Order Optimality Conditions

In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) . Finally, open problems and perspectives are also discussed.     

Applications of time scale symplectic systems without normality

Recently, the authors obtained new characterizations of the positivity and nonnegativity of a time scale quadratic functional F with separable endpoints related to a time scale symplectic system (S). In these results, the assumption of normality is absent. In this paper we present applications of such results. Namely, without assuming normality we derive Sturmian comparison theorems, results for general jointly varying endpoints, and characterizations of the positivity of F via the corresponding time scale Riccati equation, a certain perturbed quadratic functional, and a time scale Riccati inequality. These results generalize and unify many recent as well as classical ones.     

Second Order Sufficiency Criteria for a Discrete Optimal Control Problem

In this work, we derive second order necessary and sufficient optimality conditions for a discrete optimal control problem with one variable endpoint and the other fixed, and with equality control constraints. In particular, the positivity of the second variation, which is a discrete quadratic functional with appropriate boundary conditions, is characterized in terms of the nonexistence of intervals conjugate to 0, the existence of a certain conjoined basis of the associated linear Hamiltonian difference system, or the existence of a symmetric solution to the implicit and explicit Riccati matrix equations. Some results require a certain minimal normality assumption, and are derived using the sensitivity analysis technique.     

一类具幂指积系数微分算子谱的离散性

应用算子直和分解法和二次型比较的方法,研究了一类具幂指积系数微分算子谱的离散性,得到了该类微分算子的谱是离散的一些充分条件.     

Time scale symplectic systems without normality

We present a theory of the definiteness (nonnegativity and positivity) of a quadratic functional F over a bounded time scale. The results are given in terms of a time scale symplectic system (S), which is a time scale linear system that generalizes and unifies the linear Hamiltonian differential system and discrete symplectic system. The novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of F, and in establishing equivalent conditions for the nonnegativity of F without any normality assumption. To reach this goal, a new notion of generalized focal points for conjoined bases (X,U) of (S) is introduced, results on the piecewise-constant kernel of X(t) are obtained, and various Picone-type identities are derived under the piecewise-constant kernel condition. The results of this paper generalize and unify recent ones in each of the discrete and continuous time setting, and constitute a keystone for further development in this theory.     

Jacobi's condition for singular extremals: An extended notion of conjugate points

For the simple fixed endpoint problem in the calculus of variations, Jacobi's condition (“there are no conjugate points in the interior of the underlying time interval”) is necessary for optimality if the trajectory under consideration is nonsingular. In this paper, we extend the notion of conjugate points so that the above condition (in terms of this new notion) is necessary also for singular extremals. This is achieved by showing that, without any additional assumption on the trajectory, the nonnegativity of the second variation on the space of admissible variations is equivalent to the nonexistence of these “extended conjugate points”.     

On the definity of quadratic forms subject to linear constraints

A short proof is given of the necessary and sufficient conditions for the positivity and nonnegativity of a quadratic form subject to linear constraints.     

Positivity of Quadratic Functionals on Time Scales: Necessity

In this work we establish that disconjugacy of a linear Hamiltonian system on time scales is a necessary condition for the positivity of the corresponding quadratic functional. We employ a certain minimal normality (controllability) assumption. Hence, the open problems stated by the author in [17], [18] are solved with the result that positivity of the quadratic functional is equivalent to disconjugacy of the Hamiltonian system on the interval under consideration. The general approach on time scales 𝕋 involves, as special cases, the well–known continuous case for 𝕋 = ℝ and recently developed discrete one for 𝕋 = ℤ so that they are unified. As applications, Sturmian type separation and comparison theorems on time scales are also provided.     

Picone type identities and definiteness of quadratic functionals on time scales

In this paper we derive a new sufficient condition for the nonnegativity of time scale quadratic functionals associated to time scale symplectic systems. To establish this result, a new global Picone formula is derived. Another proof of a special case of the result is shown to be obtained via a Sturmian comparison technique. Furthermore, we derive several new Picone type identities which, in particular, do not impose a certain delta-differentiability assumption, and we survey known ones from the literature. The results in this paper complete our earlier work on the definiteness of a time scale quadratic functional in terms of its corresponding time scale symplectic system.     

Oscillation theorems and Rayleigh principle for linear Hamiltonian and symplectic systems with general boundary conditions

The aim of this paper is to establish the oscillation theorems, Rayleigh principle, and coercivity results for linear Hamiltonian and symplectic systems with general boundary conditions, i.e., for the case of separated and jointly varying endpoints, and with no controllability (normality) and strong observability assumptions. Our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with Dirichlet boundary conditions, and the problem with jointly varying endpoints into a problem with separated endpoints. These more general results on time scales then provide new results for the continuous time linear Hamiltonian systems as well as for the discrete symplectic systems. This paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions. Furthermore, the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories.     

Difference schemes on uniform grids performed by general discrete operators

Our aim is to set the foundations of a discrete vectorial calculus on uniform n-dimensional grids, that can be easily reformulated on general irregular grids. As a key tool we first introduce the notion of tangent space to any grid node. Then we define the concepts of vector field, field of matrices and inner products on the space of grid functions and on the space of vector fields, mimicking the continuous setting. This allows us to obtain the discrete analogous of the basic first order differential operators, gradient and divergence, whose composition define the fundamental second order difference operator. As an application, we show that all difference schemes, with constant coefficients, for first and second order differential operators with constant coefficients can be seen as difference operators of the form for suitable choices of q, and  . In addition, we characterize special properties of the difference scheme, such as consistency, symmetry and positivity in terms of q, and  .     

Sensitivity analysis for discrete optimal control problems

We present local sensitivity analysis for discrete optimal control problems with varying endpoints in the case when the customary regularity of boundary conditions can be violated. We study the behavior of the optimal solutions subject to parametric perturbations of the problem.     

Sufficient conditions for variational problems with variable endpoints: Coupled points

In this paper we demonstrate that the notion of coupled points developed in [29] for the variable endpoints variational problems is the analog of that of conjugate points when the endpoints are fixed. We provide weak and strong local optimality criteria using the strengthening of necessary conditions involving both the coupled points and the regularity concepts.This research was supported by a grant from NSERC Canada and a summer support from Michigan State University.     

Riccati equations for abnormal time scale quadratic functionals

This paper focuses on developing new Riccati type conditions for an abnormal time scale symplectic system (S). These conditions provide characterizations of the nonnegativity (with and without a certain “image condition”) and positivity of the quadratic functionals associated with such a system. The novelty of these conditions rely on the natural conjoined basis (Xa,Ua) of (S) in which Xa(t) is not necessarily invertible, and thus the system (S) could be abnormal. These results are new even in the special case of continuous time, as are some of them in the discrete time setting.     

The use of first integrals in the problem of optimal control of a linear system with a quadratic functional

We propose a method for constructing an optimal control of a linear system in a variational problem with fixed time for the control process, fixed endpoints of the phase trajectory, and a quadratic functional. The method is based on the use of first integrals of the equations of unperturbed motion. We obtain sufficient conditions for complete controllability of the linear nonstationary system.     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.