Showing papers in "Siam Journal on Control and Optimization in 2002"

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

Abstract: In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.

Robust Filtering of Discrete-Time Linear Systems with Parameter Dependent Lyapunov Functions

Abstract: Robust filtering of linear time-invariant discrete-time uncertain systems is investigated through a new parameter dependent Lyapunov matrix procedure. Its main interest relies on the fact that the Lyapunov matrix used in stability checking does not appear in any multiplicative term with the uncertain matrices of the dynamic model. We show how to use such an approach to determine high performance H2 robust filters by solving a linear problem constrained by linear matrix inequalities (LMIs). The results encompass the previous works in the quadratic Lyapunov setting. Numerical examples illustrate the theoretical results.

Nonlinear Observer Design in the Siegel Domain

Abstract: We extend the method of Kazantzis and Kravaris [Systems Control Lett., 34 (1998), pp. 241--247] for the design of an observer to a larger class of nonlinear systems. The extended method is applicable to any real analytic observable nonlinear system. It is based on the solution of a first-order, singular, nonlinear PDE. This solution yields a change of state coordinates which linearizes the error dynamics. Under very general conditions, the existence and uniqueness of the solution is proved. Lyapunov's auxiliary theorem and Siegel's theorem are obtained as corollaries. The technique is constructive and yields a method for constructing approximate solutions.

On the Modelling and Stabilization of Flows in Networks of Open Canals

Abstract: In this paper, we present a model for the controlled flow of a fluid through a network of canals using a coupled system of St. Venant equations. We then generalize in a variety of ways recent results of Coron, d'Andrea-Novel, and Bastin concerning the stabilizability around equilibrium of the flow through a single channel. This work is based on the theory of quasilinear hyperbolic systems and, in particular, on a delicate result of Li Ta-tsien concerning the existence and decay of global classical solutions.

Averaging and Vibrational Control of Mechanical Systems

Abstract: This paper investigates averaging theory and oscillatory control for a large class of mechanical systems. A link between averaging and controllability theory is presented by relating the key concepts of averaged potential and symmetric product. Both analysis and synthesis results are presented within a coordinate-free framework based on the theory of affine connections. The analysis focuses on characterizing the behavior of mechanical systems forced by high amplitude high frequency inputs. The averaged system is shown to be an affine connection system subject to an appropriate forcing term. If the input codistribution is integrable, the subclass of systems with Hamiltonian equal to "kinetic plus potential energy" is closed under the operation of averaging. This result precisely characterizes when the notion of averaged potential arises and how it is related to the symmetric product of control vector fields. Finally, a notion of vibrational stabilization for mechanical systems is introduced, and sufficient conditions are provided in the form of linear matrix equality and inequality tests.

A Leader-Follower Stochastic Linear Quadratic Differential Game

Abstract: A leader-follower stochastic differential game is considered with the state equation being a linear Ito-type stochastic differential equation and the cost functionals being quadratic. We allow that the coefficients of the system and those of the cost functionals are random, the controls enter the diffusion of the state equation, and the weight matrices for the controls in the cost functionals are not necessarily positive definite. The so-called open-loop strategies are considered only. Thus, the follower first solves a stochastic linear quadratic (LQ) optimal control problem with the aid of a stochastic Riccati equation. Then the leader turns to solve a stochastic LQ problem for a forward-backward stochastic differential equation. If such an LQ problem is solvable, one obtains an open-loop solution to the two-person leader-follower stochastic differential game. Moreover, it is shown that the open-loop solution admits a state feedback representation if a new stochastic Riccati equation is solvable.

Exact Boundary Controllability for Quasi-Linear Hyperbolic Systems

Abstract: Using a result on the existence and uniqueness of the semiglobal C1 solution to the mixed initial-boundary value problem for first order quasi-linear hyperbolic systems with general nonlinear boundary conditions, we establish the exact boundary controllability for quasi-linear hyperbolic systems if the C1 norm of initial and final states is small enough.

On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient

Abstract: We present some results concerning the controllability of a quasi-linear parabolic equation (with linear principal part) in a bounded domain of ${\mathbb R}^N$ with Dirichlet boundary conditions. We analyze the controllability problem with distributed controls (supported on a small open subset) and boundary controls (supported on a small part of the boundary). We prove that the system is null and approximately controllable at any time if the nonlinear term $f( y, abla y)$ grows slower than $|y| \log^{3/2}(1+ |y| + | abla y|) + | abla y| \log^{1/2}(1+ |y| + | abla y|)$ at infinity (generally, in this case, in the absence of control, blow-up occurs). The proofs use global Carleman estimates, parabolic regularity, and the fixed point method.

Identification For Control: Optimal Input Design With Respect To A Worst-Case $\nu$-gap Cost Function

Abstract: Parameter identification experiments deliver an identified model together with an ellipsoidal uncertainty region in parameter space. The objective of robust controller design is thus to stabilize all plants in the identified uncertainty region. The subject of the present contribution is to design an identification experiment such that the worst-case $ u$-gap over all plants in the resulting uncertainty region between the identified plant and plants in this region is as small as possible. The experiment design is performed via input power spectrum optimization. Two cost functions are investigated, which represent different levels of trade-off between accuracy and computational complexity. It is shown that the input optimization problem with respect to these cost functions is amenable to standard numerical algorithms used in convex analysis.

Strong Optimality for a Bang-Bang Trajectory

Abstract: In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C0 topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.

A Diffusion Model for Optimal Dividend Distribution for a Company with Constraints on Risk Control

Abstract: This paper investigates a model of a corporation which faces constant liability payments and which can choose a production/business policy from an available set of control policies with different expected profits and risks. The objective is to find a business policy and a dividend distribution scheme so as to maximize the expected present value of the total dividend distributions. The main feature of this paper is that there are constraints on business activities such as inability to completely eliminate risk (even at the expense of reducing the potential profit to zero) or when such a risk cannot exceed a certain level. The case in which there is no restriction on the dividend pay-out rates is dealt with. This gives rise to a mixed regular-singular stochastic control problem. First the value function is analyzed in great detail and in particular is shown to be a viscosity solution of the corresponding Hamilton--Jacobi--Bellman (HJB) equation. Based on this it is further proved that the value function must be twice continuously differentiable. Then a delicate analysis is carried out on the HJB equation, leading to an explicit expression of the value function as well as the optimal policies.

Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions

Abstract: In this paper, we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward SDEs and forward-backward SDEs. The controlled process $(X^ u,Y^ u)$ takes values in ${\mathbb R}^d \times {\mathbb R}$ and a given initial data for $X^{ u}(0)$. Then the control problem is to find the minimal initial data for $Y^{ u}$ so that it reaches a stochastic target at a specified terminal time T. The main application is from financial mathematics, in which the process $X^{ u}$ is related to stock price, $Y^{ u}$ is the wealth process, and $ u$ is the portfolio. We introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation. The boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense. This provides a unique characterization of the value function which is the minimal initial data for $Y^{ u}$.

Stability of Planar Switched Systems: The Linear Single Input Case

Abstract: We study the stability of the origin for the dynamical system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative real part, $x\in {\mbox{{\bf R}}}^2$, and $u(.):[0,\infty[\to[0,1]$ is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.). The result is obtained without looking for a common Lyapunov function but studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameters: the first depends only on the eigenvalues of A, the second depends only on the eigenvalues of B, and the third contains the interrelation among the two systems, and it is the cross ratio of the four eigenvectors of A and B in the projective line CP1. In the space of these parameters, the shape and the convexity of the region in which there is stability are studied. This bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation.

Semiconcave Control-Lyapunov Functions and Stabilizing Feedbacks

Abstract: We study the general problem of stabilization of globally asymptotically controllable systems. We construct discontinuous feedback laws, and particularly we make it possible to choose these continuous outside a small set (closed with measure zero) of discontinuity in the case of control systems which are affine in the control; moreover this set of singularities is shown to be repulsive for the Caratheodory solutions of the closed-loop system under an additional assumption.

Pole Placement by Static Output Feedback for Generic Linear Systems

Abstract: We consider linear systems with m inputs, p outputs, and McMillan degree n such that n=mp. If both m and p are even, we show that there is a nonempty open (in the usual topology) subset U of such systems, where the real pole placement map is not surjective. It follows that, for each system in U, there exists an open set of pole configurations, symmetric with respect to the real line, which cannot be assigned by any real static output feedback.

Immersion of Nonlinear Systems into Linear Systems Modulo Output Injection

Abstract: The problem of the immersion of a SISO system into a linear up to an output injection one is studied in order to design Luenberger-like observers. Necessary and sufficient conditions are stated within a very general framework. Effective computations and examples are then provided.

Indirect Boundary Stabilization of Weakly Coupled Hyperbolic Systems

Abstract: This work is concerned with the boundary stabilization of an abstract system of two coupled second order evolution equations wherein only one of the equations is stabilized (indirect damping; see, e.g., J. Math. Anal. Appl., 173 (1993), pp. 339--358). We show that, under a condition on the operators of each equation and on the boundary feedback operator, the energy of smooth solutions of this system decays polynomially at $\infty$. We then apply this abstract result to several systems of partial differential equations (wave-wave systems, Kirchhoff--Petrowsky systems, and wave-Petrowsky systems).

Continuous-Time Dynkin Games with Mixed Strategies

Abstract: Let (X,Y,Z) be a triple of payoff processes defining a Dynkin game \tilde R(\sigma,\tau) &=& E\left[ X_\sigma\1_{\{\tau>\sigma\}} +Y_\tau \1_{\{\tau<\sigma\}} +Z_\tau \1_{\{\tau=\sigma\}}\right] , where $\sigma$ and $\tau$ are stopping times valued in [0,T]. In the case Z=Y, it is well known that the condition X $\leq$ Y is needed in order to establish the existence of value for the game, i.e., $\inf_{\tau}\sup_{\sigma}\tilde R(\sigma,\tau)$ $=$ $\sup_{\sigma}\inf_{\tau}\tilde R(\sigma,\tau)$. In order to remove the condition $X$ $\leq$ $Y$, we introduce an extension of the Dynkin game by allowing for an extended set of strategies, namely, the set of mixed strategies. The main result of the paper is that the extended Dynkin game has a value when $Z\leq Y$, and the processes X and Y are restricted to be semimartingales continuous at the terminal time T.

Optimal Strategies for Risk-Sensitive Portfolio Optimization Problems for General Factor Models

Abstract: We consider constructing optimal strategies for risk-sensitive portfolio optimization problems on an infinite time horizon for general factor models, where the mean returns and the volatilities of individual securities or asset categories are explicitly affected by economic factors. The factors are assumed to be general diffusion processes. In studying the ergodic type Bellman equations of the risk-sensitive portfolio optimization problems, we introduce some auxiliary classical stochastic control problems with the same Bellman equations as the original ones. We show that the optimal diffusion processes of the problem are ergodic and that under some condition related to integrability by the invariant measures of the diffusion processes we can construct optimal strategies for the original problems by using the solution of the Bellman equations.

A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms

Abstract: We present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term. The sensitivity analysis is based on shift-variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations. Based on a first order approximation by shift-variations in L1 we introduce the concept of shift-differentiability, which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking-type functionals. In the main part of the paper we show that entropy solutions are generically shift-differentiable at almost all times t>0 with respect to the control. Hereby we admit shift-variations for the initial data which allows us to use the shift-differentiability result repeatedly over time slabs. This is useful for the design of optimization methods with time domain decomposition. Our analysis, especially of the shock sensitivity, combines structural results by using generalized characteristics and an adjoint argument. Our adjoint-based shock sensitivity analysis does not require us to restrict the richness of the shock structure a priori and admits shock generation points. The analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one-sided Lipschitz condition. As a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking-type functionals.

Geometric Description of Vakonomic and Nonholonomic Dynamics. Comparison of Solutions

Abstract: We treat the vakonomic dynamics with general constraints within a new geometric framework, which can be useful in the study of optimal control problems. We compare our formulation with the one of Vershik and Gershkovich in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description which allows us to develop an algorithm of comparison between the solutions of both dynamics. Examples illustrating the theory are treated.

Second Order Sufficient Conditions for Optimal Control Problems with Free Final Time: The Riccati Approach

Abstract: Second order sufficient conditions (SSC) for control problems with control-state constraints and free final time are presented. Instead of deriving such SSC from first principles, we transform the control problem with free final time into an augmented control problem with fixed final time for which well-known SSC exist. SSC are then expressed as a condition on the positive definiteness of the second variation. A convenient numerical tool for verifying this condition is based on the Riccati approach, where one has to find a bounded solution of an associated Riccati equation satisfying specific boundary conditions. The augmented Riccati equations for the augmented control problem are derived, and their modifications on the boundary of the control-state constraint are discussed. Two numerical examples, (1) the classical Earth-Mars orbit transfer in minimal time and (2) the Rayleigh problem in electrical engineering, demonstrate that the Riccati equation approach provides a viable numerical test of SSC.

The Topological Asymptotic Expansion for the Dirichlet Problem

Abstract: The topological sensitivity analysis provides an asymptotic expansion of a shape function when creating a small hole inside a domain. This expansion yields a descent direction which can be used for shape optimization if one wishes to keep a classical domain throughout the optimization process. In this paper, such an expansion is obtained for the Poisson equation for a large class of cost functions and arbitrarily shaped holes. In the three-dimensional case, this expansion depends on the shape of the hole but not on its orientation if the cost function involves only the solution u to the underlying partial differential equation, whereas it may also depend on its orientation if the cost function involves the gradient $ abla u$. In contrast, the asymptotic expansion is independent of the shape in the two-dimensional case. A numerical example illustrates the use of the asymptotic expansion, which yields a minimizing sequence of classical domains in a case where no classical solution exists.

On Stability of Bang-Bang Type Controls

Abstract: From the theory of nonlinear optimal control problems it is known that the solution stability w.r. t. data perturbations and conditions for strict local optimality are closely related facts. For important classes of control problems, sufficient optimality conditions can be formulated as a combination of the independence of active constraints' gradients and certain coercivity criteria. In the case of discontinuous controls, however, common pointwise coercivity approaches may fail. In the paper, we consider sufficient optimality conditions for strong local minimizers which make use of an integrated Hamilton--Jacobi inequality. In the case of linear system dynamics, we show that the solution stability (including the switching points localization) is ensured under relatively mild regularity assumptions on the switching function zeros. For the objective functional, local quadratic growth estimates in L1 sense are provided. An example illustrates stability as well as instability effects in case the regularity condition is violated.

Multidimensional Backward Stochastic Riccati Equations and Applications

Abstract: Backward stochastic Riccati differential equations (BSRDEs for short) arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients. The existence and uniqueness question of the global adapted solutions has been open since Bismut's pioneering research publication in 1978 [Seminaire de Probabilites XII, Lecture Notes in Math. 649, C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer--Verlag, Berlin, 1978, pp. 180--264]. One distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component. In a previous article [Stochastic Process. Appl., 97 (2002), pp. 255--288], the authors solved the one-dimensional case driven by Brownian motions. In this paper the multidimensional case driven by Brownian motions is studied. A closeness property for solutions of BSRDEs with respect to their coefficients is stated and is proved for general BSRDEs, which is used to obtain the existence of a global adapted solution to some BSRDEs. The global existence and uniqueness results are obtained for two classes of BSRDEs, whose generators contain a quadratic term of L (the second unknown component). More specifically, the two classes of BSRDEs are (for the regular case N >>0) @{}[email protected]{ }[email protected]{ }[email protected]{}} dK &=&\displaystyle -[A^*K+KA+Q-LD(N+D^*KD)^{-1}D^*L]\, dt +L\, dw,\\ K(T)&=&M,\end{array}\right. under the condition d=1$, and (for the singular case) $$\left\{\begin{array}{@{}[email protected]{ }[email protected]{ }[email protected]{}} dK &=&\displaystyle -[A^*K+KA+C^*KC+Q+C^*L+LC \\ && \displaystyle-(KB+C^*KD+LD) (D^*KD)^{-1}(KB+C^*KD+LD)^*]\, dt \displaystyle+L\, dw,\\ K(T)&=&M,\end{array}\right. $$ under the condition d=1 and m=n. The arguments given in this paper are completely new, and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above. We make full use of the special structure (the nonnegativity of the quadratic term, for example) of the underlying Riccati equation. Applications in optimal stochastic control are exposed.

On Reachability Under Uncertainty

Abstract: The paper studies the problem of reachability for linear systems in the presence of uncertain (unknown but bounded) input disturbances that may also be interpreted as the action of an adversary in a game-theoretic setting. It defines possible notions of reachability under uncertainty emphasizing the differences between reachability under open-loop and closed-loop control. Solution schemes for calculating reachability sets are then indicated. The situation when observations arrive at given isolated instances of time leads to problems of anticipative (maxmin) or nonanticipative (minmax) piecewise open-loop control with corrections and to the respective notions of reachability. As the number of corrections tends to infinity, one comes in both cases to reachability under nonanticipative feedback control. It is shown that the closed-loop reach sets under uncertainty may be found through a solution of the forward Hamilton--Jacobi--Bellman--Isaacs (HJBI) equation. The basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control.

On the $\lambda$-Equations for Matching Control Laws

Abstract: We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely characterized by a linear system of first order partial differential equations for one set of variables ($\,\lambda$) followed by a linear system of first order partial differential equations for the second set of variables ($\,{\widehat g}$, $\,{\widehat V}$). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the $\,\lambda$-equations. We give four examples illustrating different features of matching control laws. The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case.

On the Observability and Detectability of Continuous-Time Markov Jump Linear Systems

Abstract: The paper introduces a new detectability concept for continuous-time Markov jump linear systems with finite Markov space that generalizes previous concepts found in the literature. The detectability in the weak sense is characterized as mean square detectability of a certain related stochastic system, making both detectability senses directly comparable. The concept can also ensure that the solution of the coupled algebraic Riccati equation associated to the quadratic control problem is unique and stabilizing, making other concepts redundant. The paper also obtains a set of matrices that plays the role of the observability matrix for deterministic linear systems, and it allows geometric and qualitative properties. Tests for weak observability and detectability of a system are provided, the first consisting of a simple rank test, similar to the usual observability test for deterministic linear systems.

Identifiability and Well-Posedness of Shaping-Filter Parameterizations: A Global Analysis Approach

Abstract: In this paper, we study the well-posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients, covariance lags, or Markov parameters. For example, we determine whether there exists a shaping filter with a prescribed window of Markov parameters and a prescribed window of covariance lags. We show that several such problems are well-posed in the sense of Hadamard; that is, one can prove existence, uniqueness (identifiability), and continuous dependence of the model on the measurements. Our starting point is the global analysis of linear systems, where one studies an entire class of systems or models as a whole, and where one views measurements, such as covariance lags and cepstral coefficients or Markov parameters, from data as functions on the entire class. This enables one to pose such problems in a way that tools from calculus, optimization, geometry, and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm-independent fashion. In this language, we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona fide coordinate system on the space of shaping filters, thereby establishing existence, uniqueness, and smooth dependence of the model parameters on the measurements from data.

Escape Function Conditions for the Observation, Control, and Stabilization of the Wave Equation

Abstract: For the linear wave equation with time-invariant coefficients on a connected compact Riemannian manifold $(\Omega,g)$ with $C^{3}$ boundary, the geodesics condition of Bardos, Lebeau, and Rauch [SIAM J. Control Optim., 30 (1992), pp. 1024--1065] is characterized in terms of escape functions, which are some Lyapunov functions on the phase space $S^{*}\bar{\Omega}$ (the unit sphere cotangent bundle). Differentiable escape functions yield a sufficient condition which is slightly less sharp but does not refer to geodesics. The escape function condition yields a straightforward geometric proof that the geodesics condition holds in the situations where first order differential multiplier methods apply. Using microlocal control results, it allows us to generalize some control results (that were obtained by multiplier methods) to variable coefficients and lower order terms. It also allows us to prove, in some class of simple situations (e.g., in $\mathbb{R}^{2}$ with constant coefficients), that no first order differential multiplier method can reach the optimal control time or control regions.