Showing papers in "Siam Journal on Control and Optimization in 1991"

Martingale and duality methods for utility maximization in a incomplete market

Abstract: The problem of maximizing the expected utility from terminal wealth is well understood in the context of a complete financial market. This paper studies the same problem in an incomplete market containing a bond and a finite number of stocks whose prices are driven by a multidimensional Brownian motion process W. The coefficients of the bond and stock processes are adapted to the filtration (history) of W, and incompleteness arises when the number of stocks is strictly smaller than the dimension of W. It is shown that there is a way to complete the market by introducing additional “fictitious” stocks so that the optimal portfolio for the thus completed market coincides with the optimal portfolio for the original incomplete market. The notion of a “least favorable” completion is introduced and is shown to be closely related to the existence question for an optimal portfolio in the incomplete market. This notion is expounded upon using martingale techniques; several equivalent characterizations are provided...

On the convergence of the proximal point algorithm for convex minimization

Abstract: The proximal point algorithm (PPA) for the convex minimization problem $\min _{x \in H} f(x)$, where $f:H \to R \cup \{ \infty \} $ is a proper, lower semicontinuous (lsc) function in a Hilbert space H is considered. Under this minimal assumption on f, it is proved that the PPA, with positive parameters $\{ \lambda _k \} _{k = 1}^\infty $, converges in general if and only if $\sigma _n = \sum_{k = 1}^n {\lambda _k \to \infty } $. Global convergence rate estimates for the residual $f(x_n ) - f(u)$, where $x_n $ is the nth iterate of the PPA and $ u \in H $ is arbitrary are given. An open question of Rockafellar is settled by giving an example of a PPA for which $x_n $ converges weakly but not strongly to a minimizes of f.

Applications of splitting algorithm to decomposition in convex programming and variational inequalities

Abstract: Recently Han and Lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets. It is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by Gabay for finding a zero of the sum of two maximal monotone operators. Gabay’s convergence analysis for the splitting algorithm is sharpened, and new applications of this algorithm to variational inequalities, convex programming, and the solution of linear complementarily problems are proposed. For convex programs with a certain separable structure, a multiplier method that is closely related to the alternating direction method of multipliers of Gabay–Mercier and of Glowinski–Marrocco, but which uses both ordinary and augmented Lagrangians, is obtained.

On a convex parameter space method for linear control design of uncertain systems

Abstract: This paper presents a new procedure for continuous and discrete-time linear control systems design. It consists of the definition of a convex programming problem in the parameter space that, when solved, provides the feedback gain. One of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation, yielding solutions to problems that have raised a great deal of interest within the last few years. This is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices. In this last case, necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and, to the authors’ knowledge, this is one of the first numerical procedures able to handle and solve this interesting design problem for high-order, continuous-time or discrete-time linear models. The theory is illustrated by examples.

Recursive stochastic algorithms for global optimization in R d

Abstract: An algorithm of the form $X_{k + 1} = X_k - a_k ( abla U(X_k ) + \xi _k ) + b_k W_k $, where $U( \cdot )$ is a smooth function on $\mathbb{R}^d $, $\{ \xi _k \} $ is a sequence of $\mathbb{R}^d $-valued random variables, $\{ W_k \} $ is a sequence of independent standard d-dimensional Gaussian random variables, $a_k = {A / k}$ and $b_k = {{\sqrt B } / {\sqrt {k\log \log k} }}$ for k large, is considered. An algorithm of this type arises by adding slowly decreasing white Gaussian noise to a stochastic gradient algorithm. It is shown, under suitable conditions on $U( \cdot )$, $\{ \xi _k \} $, A, and B, that $X_k $ converges in probability to the set of global minima of $U( \cdot )$. No prior information is assumed as to what bounded region contains a global minimum. The analysis is based on the asymptotic behavior of the related diffusion process $dY(t) = - abla U(Y(t))dt + c(t)dW(t)$, where $W( \cdot )$ is a standard d-dimensional Wiener process and $c(t) = {{\sqrt C } / {\sqrt {\log t} }}$ for t large.

Duality relationships for entropy-like minimization problems

Abstract: This paper considers the minimization of a convex integral functional over the positive cone of an $L_p $ space, subject to a finite number of linear equality constraints. Such problems arise in spectral estimation, where the bjective function is often entropy-like, and in constrained approximation. The Lagrangian dual problem is finite-dimensional and unconstrained. Under a quasi-interior constraint qualification, the primal and dual values are equal, with dual attainment. Examples show the primal value may not be attained. Conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum. These conditions are satisfied in many examples.

A characterization of all solutions to the four block general distance problem

Abstract: All solutions to the four block general distance problem which arises in $H^\infty $ optimal control are characterized. The procedure is to embed the original problem in an all-pass matrix which is constructed. It is then shown that part of this all-pass matrix acts as a generator of all solutions. Special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all-pass transfer functions. As an application, necessary and sufficient conditions are found for the existence of an $H^\infty $ optimal controller. Following that, a descriptor representation of all solutions is derived.

An exact penalization viewpoint of constrained optimization

Abstract: In their seminal papers Eremin [Soviet Mathematics Doklady, 8 (1966), pp. 459–462] and Zangwill [Management Science, 13 (1967), pp. 344–358] introduce a notion of exact penalization for use in the development of algorithms for constrained optimization. Since that time, exact penalty functions have continued to play a key role in the theory of mathematical programming. In the present paper, this theory is unified by showing how the Eremin–Zangwill exact penalty functions can be used to develop the foundations of the theory of constrained optimization for finite dimensions in an elementary and straightforward way. Regularity conditions, multiplier rules, second-order optimality conditions, and convex programming are all given interpretations relative to the Eremin–Zangwill exact penalty functions. In conclusion, a historical review of those results associated with the existence of an exact penalty parameter is provided.

Sufficient conditions for dynamic state feedback linearization

Abstract: Sufficient conditions are given for the existence of a dynamic state feedback compensator for a multi-input nonlinear system such that the closed loop system is transformable into a linear controllable one by an extended state space change of coordinates An example shows that the conditions are not necessary Necessary conditions are also given which are shown to be sufficient when the number of states minus the number of controls is equal to one Several examples illustrate how the sufficient conditions obtained lead to the design of the dynamic compensator

H ∞ control of linear time-varying systems: a state-space approach

Abstract: In this paper the standard problem of $H^\infty $ control theory for finite-dimensional linear time-varying continuous-time plants is considered. The problem is: given a real number $\gamma > 0$, f...

H ∞ control with transients

Abstract: In $H_\infty $ (or uniformly optimal) control problems, it is usually assumed that the system initial conditions are zero. In this paper, an $H_\infty $-like control problem that incorporates uncertainty in initial conditions is formulated. This is done by defining a worst-case performance measure. Both finite and infinite horizon problems are considered. Necessary and sufficient conditions are derived for the existence of controllers that yield a closed-loop system for which the above-mentioned performance measure is less than a prespecified value. State-space formulae for the controllers are also presented.

Balanced parametrization of classes of linear systems

Abstract: Canonical forms and parametrizations are presented for several sets of minimal systems of given dimension: asymptotically stable systems, allpass systems, bounded real systems, positive real systems, minimum-phase systems, and the class of all minimal systems. The approach is based on balancing techniques for these classes of systems. Applications of these results to Hankel operators and model reduction are discussed.

Calmness and exact penalization

Abstract: The notion of calmness, which was introduced by Clarke and Rockafellar for constrained optimization, is considered. An equivalence to the technique of exact penalization due to Eremin and Zangwill is established. It is then shown that calmness is satisfied on a dense subset of the domain of the optimal value function.

Some characterizations of optimal trajectories in control theory

Abstract: Several characterizations of optimal trajectories for the classical Mayer problem in optimal control are provided. For this purpose the regularity of directional derivatives of the value function is studied: for instance, it is shown that for smooth control systems the value function V is continuously differentiable along an optimal trajectory $x:[t_0 ,1] \to {\bf R}^n $ provided V is differentiable at the initial point $(t_0 ,x(t_0 ))$.Then the upper semicontinuity of the optimal feedback map is deduced. The problem of optimal design is addressed, obtaining sufficient conditions for optimality. Finally, it is shown that the optimal control problem may be reduced to a viability one.

On subdifferential of optimal value functions

Abstract: A general mathematical programming problem in which the constraints are defined by multifunctions and depend on a parameter u, and the resulting value function $m(u)$ are considered. In the context...

Rapid boundary stabilization of the wave equation

Abstract: The boundary stabilization of the wave equation in bounded domains is studied. It is shown that a particular choice of the feedback leads to fast energy decay. Explicit decay rate estimates are obt...

The singular control problem with dynamic measurement feedback

Abstract: This paper is concerned with the $H_\infty $ problem with measurement feedback. The problem is to find a dynamic feedback from the measured output to the control input such that the closed-loop system has an $H_\infty $ norm strictly less than some a priori given bound $\gamma $ and such that the closed-loop system is internally stable. Necessary and sufficient conditions are given under which such a feedback exists. The only assumption that must be made is that there are no invariant zeros on the imaginary axis for two subsystems. Contrary to recent publications no assumptions are made on the direct feedthrough matrices of the plant. It turns out that this problem can be reduced to an almost disturbance decoupling problem with measurement feedback and internal stability, i.e., the problem in which we can make the $H_\infty $ norm arbitrarily small.

Approximation theory for linear-quadratic-Guassian optimal control of flexible structures

Abstract: This paper presents approximation theory for the linear-quadratic-Gaussian optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite-dimensional compensators that approximate closely the optimal compensator, which is infinite-dimensional. Design of the optimal compensator separates into an optimal linear-quadratic control problem and a dual optimal state estimation problem; the solution to each problem lies in the solution to an infinite-dimensional Riccati operator equation. The approximation scheme in the paper approximates the infinite-dimensional LQG problem with a sequence of finite-dimensional LQG problems defined for a sequence of finite-dimensional, usually finite-element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem.The finite-dimensional equations for numerical approximation ...

Necessary conditions for optimal control of distributed parameter systems

Abstract: The Ekeland variational principle is used to prove a maximum principle for the optimal controls of a general semilinear evolutionary control system in a Banach space with strictly convex dual. The boundary constraint is of general type which includes the optimal periodic control problem as a special case.

A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems

Abstract: A hybrid method for computing the feedback gains in linear quadratic regulator problems is proposed. The method, which combines use of a Chandrasekhar type system with an iteration of the Newton–Kleinman form with variable acceleration parameter Smith schemes, is formulated to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropirate when used with large dimensional systems such as those arising in approximating infinite-dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantages of the proposed algorithm over the standard eigenvector (Potter, Laub–Schur) based techniques are discussed, and numerical evidence of the efficacy of these ideas is presented.

On constructing nonlinear observers

Abstract: Two algorithmic approaches to constructing nonlinear observers currently exist. Here one of these is improved by finding an explicit solution to the partial differential equations describing the change of state coordinates, thereby avoiding expensive bracket computations. This simplifies the algorithm and has implications for system identification.

Numerical Methods for Stochastic Singular Control Problems

Abstract: The paper develops a powerful class of numerical methods for stochastic singular control problems. The basic models used are diffusion or reflected diffusions, but the method is of general applicability. The central idea is that of the Markov chain approximation method, where an approximation to the control problem is found for which an optimal solution is computable, and which is an arbitrarily good approximation to the original problem and its optimal value function. The methods are convenient to program and use (and they have been used with success), and they cover a wide variety of problems. In fact, for the singular problem, they seem to be the only ones currently available. Owing to problems in proving tightness of certain processes that occur in the convergence proofs, the methods of proof used for the nonsingular problems need modifications. Examples of useful approximations, the algorithms, and the convergence proofs are given. To illustrate the power of the methods, two classes of problems are d...

Markov decision problems and state-action frequencies

Abstract: Consider a controlled Markov chain with countable state and action spaces. Basic quantities that determine the values of average cost functionals are identified. Under some regularity conditions, these turn out to be a collection of numbers, one for each state-action pair, describing for each state the relative number of uses of each action. These "conditional frequencies," which are defined pathwise, are shown to determine the "state-action frequencies" that, in the finite case, are known to determine the costs. This is extended to the countable case, allowing for unbounded costs. The space of frequencies is shown to be compact and convex, and the extreme points are identified with stationary deterministic policies. Conditions under which the search for optimality in several optimization problems may be restricted to stationary policies are given. These problems include the standard Markov decision process, as well as constrained optimization (both in terms of average cost functionals) and variability-sensitive optimization. An application to a queueing problem is given, where these results imply the existence and explicit computation of optimal policies in constrained optimization problems. The pathwise definition of the conditional frequencies implies that their values can be controlled directly; moreover, they depend only on the limiting behavior of the control. This has immediate application to adaptive control of Markov chains, including adaptive control under constraints.

Initial boundary value problems and optimal control for nonautonomous parabolic systems

Abstract: A large class of linear nonautonomous parabolic systems in bounded domains is considered, with control acting on the boundary through Dirichlet or Neumann conditions, from the point of view of semigroup theory. The results from [Rend. Sem. Mat. Univ. Padova, 78 (1987), pp. 47–107], [On fundamental solutions for abstract parabolic equations, Lecture Notes in Math., Vol. 1223, Springer-Verlag, Berlin, Heidelberg, 1986, pp. 1–11] on abstract homogeneous parabolic Cauchy problems allow operators with varying domains and Holder continuous coefficients to be handled. A representation formula for solutions corresponding to square integrable control functions is derived and used to solve a linear-quadratic regulator problem over finite time horizon, by a direct study of the associated integral Riccati equation.

Second-order Hamilton-Jacobi equations in infinite dimensions

Abstract: Some second-order Hamilton–Jacobi equations connected to stochastic optimal control problems for infinite-dimensional systems driven by a white noise are studied. A direct method to prove existence and uniqueness of mild solutions is developed. Then this solution is identified as the value function of the related stochastic control problem, and a feedback formula for optimal controls is derived.

Velocity method and Lagrangian formulation for the computation of the shape Hessian

Abstract: The object of this paper is to study the shape Hessian of a shape functional by the velocity (speed) method. It contains a review and an extension of the velocity method and its connections with methods using first- or second-order perturbations of the identity. The key point is that all these methods yield the same shape gradient but different and unequal shape Hessian since each method depends on a choice of “connection.” However, for autonomous velocity fields the velocity method yields a canonical bilinear Hessian. Expressions obtained by other methods can be recovered by adding to that canonical term the shape gradient acting on the acceleration of the velocity field associated with the choice of perturbation of the identity. The second part of the paper is an application of the Lagrangian method with function space embedding to compute the shape gradient and Hessian of a simple cost function associated with the nonhomogeneous Dirichlet problem.

Existence of control Lyapunov functions and application to state feedback stabilizabilty of nonlinear systems

Abstract: The asymptotic and practical stabilization for the affine in the control nonlinear systems, which extends the results of Artstein, Sontag, and Tsinias is explored. Sufficient conditions for the existence of control Lyapunov functions are presented guaranteeing stabilization. The corresponding feedback laws are smooth, except possibly at the equilibrium of the system.

On the convergence of a matrix splitting algorithm for the symmetric monotone linear complementarity problem

Abstract: A matrix splitting algorithm for the linear complementarily problem is considered, where the matrix is symmetric positive semidefinite. It is shown that if the splitting is regular, then the iterates generated by the algorithm are well defined and converge to a solution. This result resolves in the affirmative a long standing question about the convergence of the point successive overrelaxation (SOR) method for solving this problem. This result is also extended to related iterative methods. As direct consequences, convergence of the methods of, respectively, Aganagic, Cottle et al., Mangasarian, Pang, and others, is obtained, without making any additional assumptions on the problem.

Model matching and factorization for nonlinear systems: a structural approach

Abstract: The model matching and the left factorization problems for nonlinear systems are investigated using an approach based on the structural algorithm. Sufficient conditions for the solvability of the f...

Feedback equivalence for nonlinear systems and the time optimal control problem

Abstract: This article relates the classification of affine control systems under the action of the feedback group, with a differential classification of a set of constrained Hamiltonian vector fields, arising from Pontryagin’s Maximum Principle, for the time minimal control problem. They represent the singularities of the input-state mapping. This relation provides a method to compute feedback invariants.