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The paper studies the open-loop saddle point and the open-loop lower and upper values, as well as their relationship for two-person zero-sum stochastic linear-quadratic (LQ, for short) differential games with deterministic coefficients. It derives a necessary condition for the finiteness of the open-loop lower and upper values and a sufficient condition for the existence of an open-loop saddle point. It turns out that under the sufficient condition, a strongly regular solution to the associated Riccati equation uniquely exists, in terms of which a closed-loop representation is further established for the open-loop saddle point. Examples are presented to show that the finiteness of the open-loop lower and upper values does not ensure the existence of an open-loop saddle point in general. But for the classical deterministic LQ game, these two issues are equivalent and both imply the solvability of the Riccati equation, for which an explicit representation of the solution is obtained., Comment: 26 pages

In this paper, we introduce a new infinite horizon domination-monotonicity framework. In this framework, by the method of continuation and some subtle techniques, we obtain an existence and uniquen...

A linear control system with quadratic cost functional over infinite time horizon is considered without assuming controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such kind of problems. Existence and non-existence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some of the above-mentioned problems and at the same time, the limitation of this approach is also revealed.

In this work, we propose a robust approach to design distributed controllers for unknown-but-sparse linear time-invariant systems. By leveraging modern techniques in distributed controller synthesi...

We present a constructive method to calculate the optimal feedback law for a general class of fractional systems, which may be unstable or incommensurate order. This method is based on a generaliza...

In this paper, we propose a novel equilibrium solution notion for the time-inconsistent stochastic linear-quadratic problem. This notion is called the mixed equilibrium solution, which consists of ...

This paper presents a continuous time stochastic linear quadratic (LQ) adaptive control algorithm for completely observed linear stochastic systems with unknown parameters. Based on a certainty equ...

A multilevel self-similar domination-monotonicity structure is proposed, and a kind of coupled forward-backward stochastic differential equations (FBSDEs) with such structure is proved to be unique...

We analyze the connection between turnpike behaviors and strict dissipativity properties for discrete time finite dimensional linear quadratic optimal control problems. We first use strict dissipativity as a sufficient condition for the turnpike property. Next, we characterize strict dissipativity and the newly introduced property of strict pre-dissipativity in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelties which distinguishes the results in the present paper from earlier ones on linear quadratic optimal control problems is the consideration of state and input constraints.

This paper addresses an open problem in the area of linear quadratic optimal control. We consider the regular, infinite-horizon, stability-modulo-a-subspace, indefinite linear quadratic problem under the assumption that the dynamics are stabilizable. Our result generalizes previous works dealing with the same problem in the case of controllable dynamics. We explicitly characterize the unique solution of the algebraic Riccati equation that gives the optimal cost and optimal feedback control, as well as necessary and sufficient conditions for the existence of optimal controls., Comment: 16 pages

A linear quadratic Dirichlet control problem governed by an elliptic equation posed on a possibly nonconvex polygonal domain is analyzed. Detailed regularity results are provided in classical Sobolev (Slobodetskiĭ) spaces. In particular, it is proved that in the presence of pointwise control constraints, the optimal control is continuous despite the nonconvexity of the domain.

This paper contains corrections to [R. Curtain, SIAM J. Control Optim., 49 (2011), pp. 2542--2557]. While all claims remain valid, and the proof for the linear quadratic Riccati equations is correct, this proof does not cover the positive-real and bounded-real Riccati equations. Here a correct proof is given for these cases.

Stability properties of the problem of minimizing a (nonconvex) linear-quadratic function over a Euclidean ball, known as the trust-region subproblem, are studied in this paper. We investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper/lower semicontinuity of the Karush--Kuhn--Tucker (KKT) point set map and the global solution map, explicit formulas for computing the directional derivative and the Frechet derivative of the optimal value function. Stability of the Karush--Kuhn--Tucker point set under the perturbation of the quadratic form is also studied.

In this paper we consider a state-constrained optimal control problem with boundary control, where the state constraints are imposed only in an interior subdomain. Our goal is to derive a priori error estimates for a finite element discretization with and without additional regularization. We will show that the separation of the set where the control acts and the set where the state constraints are given improves the approximation rates significantly. The theoretical results are illustrated by numerical computations.

This paper considers optimal (minimizing) control of stochastic linear quadratic regulators (LQRs). The assumption that the control weight costs must be positive definite, inherited from the determ...

The number $\gamma:=\Vert \hat Q^{-\half}\hat R\hat P^{-\half}\Vert$ is an important parameter for the extended linear-quadratic programming (ELQP) problem associated with the Lagrangian $L(\hat u,\hat v) = \hat p \mdot \hat u+ \half \hat u \mdot \hat P\hat u + \hat q\mdot \hat v - \half \hat v \mdot \hat Q\hat v - \hat v\mdot \hat R\hat u $ over polyhedral sets $\hat U \times \hat V.$ Some fundamental properties of the problem, as well as the convergence rates of certain newly developed algorithms for large-scale ELQP, are all related to $\gamma.$ In this paper, we derive an estimate of $\gamma$ for the ELQP problems resulting from discretization of an optimal control problem. We prove that the parameter $\gamma$ of the discretized problem is bounded independently of the number of subintervals in the discretization.

The aim of this paper is two-fold. First, new variants are proposed for the primal-dual steepest descent algorithm as one in the family of primal-dual projected gradient algorithms developed by Zhu and Rockafellar [SIAM J. Optim., 3 (1993), pp. 751–783] for large-scale extended linear-quadratic programming. The variants include a second update scheme for the iterates, where the primal-dual feedback is arranged in a new pattern, as well as alternatives for the “perfect line search” in the original version of the reference. Second, new linear convergence results are proved for all these variants of the algorithm, including the original version as a special case, without the additional assumptions used by Zhu and Rockafellar. For the variants with the second update scheme, a much sharper estimation for the rate of convergence is obtained due to the new primal-dual feedback pattern.

This paper describes a technique for generating both linear and linear-quadratic bilevel programming problems. The method is based on combining a number of single-parameter two-variable problems to obtain a separable multivariable problem with a number of desirable properties. The separability is then disguised using a simple transformation.The proposed technique, which requires very little computational effort, allows the user control over, among other things, the number and type of minima and the data density.

This paper studies the following form of nonlinear stochastic partial differential equation: \[ \begin{gathered} - d\Phi _t = \mathop {\inf }_{v \in U} \left\{ {\frac{1}{2}\sum_{i,j} {\left[ {\sigma \sigma ^ * } \right]_{ij} (x,v,t)} \partial _{x_i x_j } \Phi _t (x) + \sum_i {b_i (x,v,t)} \partial _{x_i } \Phi _t (x) + L(x,v,t)} \right. \hfill \\ \qquad \qquad \left. { + \sum_{i,j} {\sigma_{ij}(x,v,t)\partial _{x_i } \Psi _{j,t} (x)} } \right\}dt - \Psi _t (x)dW_t ,\quad \Phi _T (x) = h(x), \hfill \\ \end{gathered}\] where the coefficients $\sigma _{ij} $, $b_i $, L, and the final datum h may be random. The problem is to find an adapted pair $(\Phi ,\Psi )(x,t)$ uniquely solving the equation. The classical Hamilton–Jacobi–Bellman (HJB) equation can be regarded as a special case of the above problem. An existence and uniqueness theorem is obtained for the case where $\sigma $ does not contain the control variable v. An optimal control interpretation is given. The linear quadratic case is discussed as well.

For a Linear Quadratic Adaptive Stochastic Optimal Control Problem a formula for the optimal control is obtained by applying variational methods to an equivalent problem obtained from the Girsanov transformation. The formula does depend on a quantity defined through a martingale representation of the conditional remaining cost.

The sine-kernel function of the sampling series is replaced by spline functions having compact support, all built up from the B-splines $M_{n} $. The resulting generalized sampling series reduces to finite sums so that no truncation error occurs. Moreover, the approximation error generally decreases more rapidly than for the classical series when W tends to infinity, $1/W$ being the distance between the sampling points. For the kernel function $\varphi (t) = 5M_4 (t) - 4M_5 (t)$ with support $[ - \tfrac{5}{2},\tfrac{5}{2} ]$, e.g., it decreases with order $O(W^{ - 4} )$ provided the signal has a fourth order derivative; for the classical series the order is $O(W^{-4}\log W)$. Seven characteristic examples are treated in detail.

A generalized approach is taken to linear and quadratic programming in which dual as well as primal variables may be subjected to bounds, and constraints may be represented through penalties. Corresponding problem models in optimal control related to continuous-time programming are then set up and theorems on duality and the existence of solutions are derived. Optimality conditions are obtained in the form of a global saddle point property which decomposes into an instantaneous saddle point condition on the primal and dual control vectors at each time, along with an endpoint condition.

Since 1970, it has been known that open loop Nash equilibrium points exist for N-person differential games when there are integral bounds on the control functions. These results were obtained using various fixed-point theorems, and required that the duration of the game be “small”.In this paper it is assumed that the controls are constrained to take values in compact, convex subsets of $R^{q_i} $. A fixed-point theorem due to Tychonov is used, and as a result there are no restrictions on the duration of the game.

The purpose of this paper is to apply the stochastic maximum principle previously obtained by the author to the control of a linear quadratic criterion.

The concepts of detectability and stabilizability are explored for time-varying systems. We study duality, invariance under feedback, an extended version of the lemma of Lyapunov, existence of stabilizing feedback laws, linear quadratic filtering and control, and the existence of approximate canonical forms.

The coefficients of the plant are perturbed in an abstract linear quadratic problem on a Hilbert space. Many characterizations are obtained about the perturbations that give strongly convergent sequences of optimal controls. Some connections are shown to exist between this continuous dependence problem and some modes of variational convergence for sequences of convex functions, in particular Mosco’s convergence. Two applications are given to the classical linear-quadratic problem, for ordinary and elliptic partial differential equations.

Consider the linear quadratic cost control problem $\dot x = A(t)x + B(t)u$, $x(0) = x_0 $,with a cost functional $J[u] = \frac{1}{2}\int_0^T {[\langle {x,Q(t)x} \rangle + \langle {u,R(t)u} \rangle ]dt} $. Let S be a suitable space of piecewise cubic polynomials on a mesh of norm h on the interval $[0,T]$. Then it is shown that a so-called Ritz–Trefftz method for minimizing $J[ \cdot ]$ over S leads to an approximation to $J[ \cdot ]$ of order $O(h^7 )$. Further, a computable error bound can be exhibited. It is also shown that the computed pair $(\bar u,\bar x)$ converges to the optimal pair $(u^ * ,x^ * )$ with order $O(h^3 )$. Similar statements are made for piecewise polynomial approximation of arbitrary positive order.