Showing papers in "Siam Journal on Control and Optimization in 1988"
TL;DR: In this article, the authors present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls, and they show that in general case, when the controls are distributed, these systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks.
Abstract: We present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls. We show that in a general case, when the controls are distributed, stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks. We also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems.
370 citations
TL;DR: In this article, the authors studied the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first-order Hamilton-Jacobi (HJ) equation up to the boundary.
Abstract: The authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first-order Hamilton-Jacobi (HJ) equation up to the boundary. This equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary. When these solutions are equal, they can be obtained through the vanishing viscosity method. Finally, when the HJ equation has a continuous solution, it is proved to be the value function for the first exit time of the domain. It is also the vanishing viscosity limit arising, in particular, in some large deviations problems.
251 citations
TL;DR: In this article, it was shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies, and Hilbert-Schmidt and nuclear bounds on the Hankel operator errors were obtained.
Abstract: The class of linear infinite-dimensional systems with finite-dimensional inputs and outputs whose impulse response h satisfies $h \in L_1 \cap L_2 (0,\infty ;\mathbb{C}^{p \times m} )$ and induces a nuclear Hankel operator is said to be of nuclear type. For this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies. Furthermore, explicit $L_\infty$ bounds on the transfer function errors, $L_1$ and $L_2$ bounds on the impulse response errors, and Hilbert-Schmidt and nuclear bounds on the Hankel operator errors are obtained. These truncations also generate an approximating sequence to the optimal Hankel-norm approximations to the original system, and various error bounds of these approximants are deduced.
234 citations
TL;DR: Inverse stochastic control as discussed by the authors, a statistician observes a realization of a controlled stochastically-constrained process and observes a sequence of states in the process.
Abstract: Consider the following “inverse stochastic control” problem. A statistician observes a realization of a controlled stochastic process $\{ d_t ,x_t \} $ consisting of the sequence of states $x_t$, a...
176 citations
TL;DR: An algorithm is proposed for the minimization of a smooth function subject to smooth inequality constraints that, provided the current iterate is feasible and the current multiplier estimates are strictly positive, the primal component of the quasi-Newton direction is a direction of descent for the objective function.
Abstract: An algorithm is proposed for the minimization of a smooth function subject to smooth inequality constraints. Unlike sequential quadratic programming type methods, this algorithm does not involve the solution of quadratic programs, but merely that of linear systems of equations. Locally the iteration can be viewed as a perturbation of a quasi-Newton iteration on both the primal and dual variables for the solution of the equalities in the Kuhn-Tucker first order conditions of optimality. It is observed that, provided the current iterate is feasible and the current multiplier estimates are strictly positive, the primal component of the quasi-Newton direction is a direction of descent for the objective function. This fact is used to induce global convergence, without the need of a surrogate merit function. A careful “bending” of the search direction prevents any Maratos-like effect, and local superlinear convergence is proven. While the algorithm requires that an initial feasible point be available, the succe...
152 citations
TL;DR: In this article, the authors investigated linear elastic systems with damping in Hilbert spaces, where A is a positive definite unbounded linear operator and B is a closed linear operator related in various ways to $A^\alpha (\frac{1}{2} \leqq \alpha \LEqq 1)$.
Abstract: In the present paper we investigate linear elastic systems with damping $\ddot y + B\dot y + Ay = 0$ in Hilbert spaces, where A is a positive definite unbounded linear operator and B is a closed linear operator related in various ways to $A^\alpha (\frac{1}{2} \leqq \alpha \leqq 1)$. We discuss the spectral property of these systems and obtain some fundamental results for the holomorphic property and the exponential stability of the semigroups associated with these systems.
148 citations
TL;DR: In this paper, a study of differentiability properties of the optimal value function and an associated optimal solution of a parametrized nonlinear program is presented under the Mangasarian-Fromovitz constraint qualification when the corresponding vector of Lagrange multipliers is not necessarily unique.
Abstract: This paper is concerned with a study of differentiability properties of the optimal value function and an associated optimal solution of a parametrized nonlinear program. Second order analysis is presented essentially under the Mangasarian-Fromovitz constraint qualification when the corresponding vector of Lagrange multipliers is not necessarily unique. It is shown that under certain regularity conditions the optimal value function possesses second order directional derivatives and the optimal solution mapping is directionally differentiable. The results obtained are applied to an investigation of metric projections in finite-dimensional spaces.
132 citations
TL;DR: In this paper, a maximum principle for dynamic optimization problems was proved, in which the control driving the system may be impulsive and give rise to discontinuous trajectories, and the approach, which involves approximating the problem by a conventional one and using Ekeland's theorem, is new.
Abstract: A Maximum Principle is proved which governs solutions to dynamic optimization problems in which the controls driving the system may be impulsive and give rise to discontinuous trajectories. The approach, which involves approximating the problem by a conventional one and using Ekeland’s theorem, is new. It permits us to weaken very considerably the hypotheses under which Maximum Principles for such problems have previously been proved.
121 citations
TL;DR: In this article, stability and sensitivity of the efficient set in convex vector optimization are considered and sufficient conditions for the upper and lower semicontinuity of the perturbation map are obtained.
Abstract: In this paper stability and sensitivity of the efficient set in convex vector optimization are considered. The perturbation map is defined as a set-valued map. It associates with each parameter vector the set of all minimal points of the parametrized feasible set with respect to an ordering cone in the objective space. Sufficient conditions for the upper and lower semicontinuity of the perturbation map are obtained. Because of the convexity assumptions, the conditions obtained are fairly simple if compared to those in the general case. Moreover, a complete characterization of the contingent derivative of the perturbation map is obtained under some assumptions. It provides quantitative information on the behavior of the perturbation map.
113 citations
TL;DR: In this article, a spectral norm bound on the relative perturbation of the solution of the Lyapunov equation was derived, which is essentially equivalent to the Frobenius norm bound obtained from the associated Kronecker product system.
Abstract: We present an analysis of the sensitivity of the solution of the Lyapunov equation $A^ * X + XA = - W$, where A is stable This analysis leads to a spectral norm bound on the relative perturbation of the solution which is optimal for a certain class of estimates and which is essentially equivalent to the Frobenius norm bound obtained from the associated Kronecker product system The latter bound can be expressed in terms of ${\operatorname{sep}}(A^ * , - A)$ and is known to accurately reflect the sensitivity of the Lyapunov problem, but it is hard to interpret in terms of the original matrix A In contrast, the spectral norm bound which we derive is directly related to the minimal $L_2$ damping of the dynamical system $\dot z = Az$ Moreover, this dynamical link with the sensitivity problem leads to a new method of systematically investigating the norm behavior of $e^{At} $ as well as providing a wealth of information about control theoretic aspects of $\dot z = Az$, when A is the closed loop state matrix
108 citations
TL;DR: In this article, an energy decay rate is obtained for solutions of wave type equations in a bounded region in the plane of the plane whose boundary consists of a nontrapping reflecting surface and an energy absorbing surface.
Abstract: An energy decay rate is obtained for solutions of wave type equations in a bounded region in $\mathbb{R}^n $ whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface. Unlike most previous results on this subject, the results presented here are potentially valid for regions having connected boundaries.
TL;DR: In this article, the authors introduced a new theorem on the differentiability of a Min Max with respect to a parameter and showed how such a theorem can be applied to compute the material derivative in shape sensitivity analysis problems.
Abstract: The object of this paper is twofold. We introduce a new theorem on the differentiability of a Min Max with respect to a parameter and we show how such a theorem can be applied to compute the material derivative in shape sensitivity analysis problems. We consider the Min Max of a functional which is parametrized by t. We show that, under appropriate conditions, the derivative of the Min Max with respect to t is the Min Max with respect to the points solution of the Min Max problem of the derivative of the original functional with respect to t. To illustrate the use of this theorem, we apply it to the control of an elliptic equation with a nondifferentiable observation and to shape design problems.
TL;DR: In this paper, the authors present a procedure for stabilizing a class of uncertain linear systems by repeated solution of an algebraic Riccati equation, which is then used to construct a stabilizing linear control law.
Abstract: This paper presents a procedure for stabilizing a class of uncertain linear systems. The uncertain systems under consideration are described by state equations in which the input matrix depends on a matrix of uncertain parameters. This matrix of uncertain parameters may be time-varying; however, it is constrained by a bound on its induced norm.The stabilization procedure presented involves the repeated solution of an algebraic Riccati equation. When a positive definite solution to this Riccati equation is obtained, this solution is used to construct a stabilizing linear control law. Another important result contained in this paper can be stated roughly as follows: For the class of uncertain systems under consideration, if a system can be stabilized via nonlinear control, then it is also possible to stabilize the system via linear control.
TL;DR: The proposed parallel algorithm has attractive convergence properties and can be implemented as parallel algorithms for tackling definite quadratic programs, linear programs, systems of linear equations and systems of generalized nonlinear inequalities.
Abstract: A parallel algorithm is proposed in this paper for solving the problem $\min \{ q(x)|x \in C_1 \cap \cdots \cap C_m \} $ where q is an uniformly convex function and $C_i$ are closed convex sets in $R^n$. In each iteration of the method, we solve in parallel m independent subproblems, each minimizing a definite quadratic function over an individual set $C_i$. The method has attractive convergence properties and can be implemented as parallel algorithms for tackling definite quadratic programs, linear programs, systems of linear equations and systems of generalized nonlinear inequalities.
TL;DR: In this paper, the authors considered a compact parameterization of the space of linear dynamical systems and introduced base points and critical points as two algebraic-geometric objects that have significance in sensitivity and robustness studies, respectively.
Abstract: This paper studies structured uncertainty problems in feedback system design, considers a compact parameterization of the space of linear dynamical systems and introduces “base points” and “critical points” as two algebraic-geometric objects that have significance in sensitivity and robustness studies, respectively. Using the Nevanlinna–Pick interpolation theory, the author obtains a necessary and sufficient condition for simultaneous stabilization of a structured one-parameter family of plants. A recent result due to Kharitonov, on the simultaneous stability of a parameterized family of polynomials, leads to a sufficiency condition for simultaneous stabilization of a structured multiparameter family of plants. Furthermore the author considers “simultaneous pole placement” of an r-tuple of plants as a means to arbitrarily tune the natural frequencies of a multimode linear dynamical system. The concept of “nondegenerate” and “twisted” r-tuples of plants is introduced as the pole placement problem is studie...
TL;DR: The classical discrete minimax problem is considered and it was found that the classical Lagrangian of the equivalent problem has a number of important properties both in primal and dual spaces in convex as well as in nonconvex cases.
Abstract: The classical discrete minimax problem is considered. It is transformed into an equivalent problem by a monotone transformation of the initial functions. It was found that the classical Lagrangian ...
TL;DR: It is proved that dynamic state feedback decoupling with stability is achievable if the number of independent inputs is large enough to compensate the intrinsic “nondecouplability” of the system.
Abstract: In this paper we solve the minimal delay decoupling problem of linear multivariable systems. We look for feedback implementable solutions which moreover guarantee closed loop stability. We prove that dynamic state feedback decoupling with stability is achievable if the number of independent inputs is large enough to compensate the intrinsic “nondecouplability” of the system. We introduce new feedback invariants characterizing the minimal number of infinite and unstable zeros of the decoupled system.
TL;DR: It is shown that, for subsystems of bilinear systems, accessibility can be decided in polynomial time, but controllability is NP-hard.
Abstract: This article compares the difficulties of deciding controllability and accessibility. These are standard properties of control systems, but complete algebraic characterizations of controllability have proved elusive. The article shows in particular that, for subsystems of bilinear systems, accessibility can be decided in polynomial time, but controllability is NP-hard.
TL;DR: In this paper, it was shown that the adjoints of a spline-based approximation scheme for delay equations do not converge strongly and that the spline approximation does not converge at all.
Abstract: It is shown that the adjoints of a spline-based approximation scheme for delay equations do not converge strongly.
TL;DR: In this paper, a new notion of coprimeness called approximate left coprimness is introduced, and it is shown that the standard observable realization associated to the representation $Q^{ - 1} * P$ is canonical if and only if Q and P are approximately left coprime.
Abstract: This paper studies matrix fractional representation for impulse responses of a certain class of infinite-dimensional linear systems which contains, in particular, delay-differential systems. Such impulse responses are called pseudo-rational in this paper. This fractional representation is effectively used to derive concrete function space models from the abstract shift realizations. Given a fractional representation $Q^{ - 1} * P$, a standard observable realization, analogous to Fuhrmann realizations for finite-dimensional systems, is associated to it. A new notion of coprimeness called approximate left coprimeness is introduced, and it is shown that the standard observable realization associated to the representation $Q^{ - 1} * P$ is canonical if and only if Q and P are approximately left coprime. Some examples are discussed to illustrate the relationships among various coprimeness concepts that have appeared in the literature.
TL;DR: In this paper, the existence of optimal stable Markov relaxed controls for the ergodic control of multidimensional diffusions is established by direct probabilistic methods based on a characterization of a.s. limit sets of empirical measures.
Abstract: The existence of optimal stable Markov relaxed controls for the ergodic control of multidimensional diffusions is established by direct probabilistic methods based on a characterization of a.s. limit sets of empirical measures. The optimality of the above is established in the strong (i.e., almost sure) sense among all admissible controls under very general conditions.
TL;DR: In this paper, the authors give a new upper bound that requires operations that only grow polynomially in the number of random variables and show that this bound is sharp if the function is linear.
Abstract: Stochastic linear programs require the evaluation of an integral in which the integrand is itself the value of a linear program. This integration is often approximated by discrete distributions that bound the integral from above or below. A difficulty with previous upper bounds is that they generally require a number of function evaluations that grows exponentially in the number of variables. We give a new upper bound that requires operations that only grow polynomially in the number of random variables. We show that this bound is sharp if the function is linear and give computational results to illustrate its performance.
TL;DR: In this article, it was shown that globally time-optimal trajectories are finite concatenations of six bang and singular arcs with a bound on the number of switchings.
Abstract: We consider the problem of time-optimal control for systems of the form $\dot x = f(x) + g(x)u$, where f and g are smooth vector fields and admissible controls are measurable scalar functions u with values in $ - 1 \leqq u \leqq 1$. Under the assumption that f, g and $[f,g]$ are independent, and that also one of the triples $(g[f,g],[f + g,[f,g]])$ or $(g[f,g],[f - g,[f,g]])$ consists of independent vectors, we show that generically every point has a neighborhood U such that time-optimal trajectories that lie in U are concatenations of at most six bang and singular arcs. This implies that globally time-optimal trajectories are finite concatenations of bang and singular arcs with a bound on the number of switchings; in particular, time-optimal controls are piecewise smooth. Results of this type are relevant for the existence of a regular synthesis.
TL;DR: For the class of inhomogeneous Markov processes arising from simulated annealing, it was shown in this paper that the Forward equations associated with such processes exist and are positive for each state i, where $T(t) is the temperature and $u(i)$ is the energy level at i.
Abstract: For the class of inhomogeneous Markov processes arising from simulated annealing, it is shown that ${{\lim _{t \to \infty } P(X_t = i)} / {\exp ({{ - u(i)} / {T(t)}})}}$ exists and is positive for each state i, where $T(t)$ is the temperature and $u(i)$ is the energy level at i (assuming that min; $u(i) = 0$). The method used is to consider the Forward equations associated with such processes.
TL;DR: In this paper, the authors consider the problem of time-optimal control for systems of the form where f and g are smooth vector fields and admissible controls are measurable functions u with values in $ - 1 \leqq u \LEqq 1$.
Abstract: We consider the problem of time-optimal control for systems of the form $\dot x = f(x) + ug(x)$ where f and g are smooth vector fields and admissible controls are measurable functions u with values in $ - 1 \leqq u \leqq 1$. Under the assumption that f, g and $[f,g]$ are independent, we prove that generically every point has a neighborhood $\mathcal{U}$ such that bang-bang trajectories that lie in $\mathcal{U}$ and have more than 7 switchings are not time-optimal.
TL;DR: In this article, the authors studied the control of diffusions under partial observations and proved the existence of an optimal Markovian filter under mild conditions and showed that the value functions for the two problems are equal.
Abstract: This paper concerns the control of diffusions under partial observations. Part I studies the control of the signal process $dX_t = b(t,X_t ,U_t)dt + \sigma (t,X_t ,U_t )dB_t $, when the observation is $dY_t = h(t,X_t )dt + dW_t $, and when the objective is to maximize a reward function $E\{ \int _r^T k(s,X_s, U_s )ds + g(X_T )\} $. The existence of an optimal relaxed control is proved.Part II studies the separated problem and proves the existence of an optimal Markovian filter. Then, the authors compare the two problems and prove, under mild conditions, that the value functions for the two problems are equal.
TL;DR: In this article, an implementable approximation of the infinite-dimensional Kalman filter is proposed for linear distributed systems corrupted by white noise, which relies on a Galerkin type treatment of both the Riccati and filter equations, and it is shown to converge to the best linear estimate of the state for each sample path of the noise.
Abstract: In this paper an implementable approximation of the infinite-dimensional Kalman filter is proposed for linear distributed systems corrupted by white noise. It relies on a Galerkin type treatment of both the Riccati and the filter equations, and it is shown to converge to the best linear estimate of the state for each sample path of the noise. A basic tool is the study of the Riccati equation on the Hilbert space of Hilbert-Schmidt operators on the state space. Numerical results are given for a case concerning delay-differential equation.
TL;DR: In this paper, a controller degree bound was established for problems in which both columns and rows are square (problems of the first kind), and the degree bound carried over to problems of the second kind without change.
Abstract: This paper is a continuation of our work on $\mathcal{H}^\infty $-optimal control problems which may be embedded in the linear fractional configuration of Fig. 1. In two previous articles [19], [20], a controller degree bound was established for problems in which both $P_{12} (s)$ and $P_{21} (s)$ are square (problems of the first kind). If the McMillan degree of $P(s)$ is n, it was shown that there exist $\mathcal{H}^\infty $-optimal controllers with McMillan degree no greater than $n - 1$.Here we switch our attention to problems of the second kind. That is, we allow $P_{12} (s)$ to have more rows than columns (with $P_{21} (s)$ square), or alternatively, we allow $P_{21} (s)$ to have more columns than rows (with $P_{12} (s)$ square). Our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change. In addition to the controller degree bound, our analysis suggests a number of modifications which are easily made to cur...
TL;DR: In this article, the authors introduce the notion of normality and conjugate points for optimal control problems and obtain new second-order necessary conditions for optimality, namely, the Jacobi condition and the existence of a solution to a Riccati equation.
Abstract: In this paper we introduce a definition of “normality” and of “conjugate points” for a general optimal control problem. Using these concepts we obtain new second-order necessary conditions for optimality. In the special case when the control set U is the whole space or in the classical setting of calculus of variations, our conditions reduce to known results, namely, the Jacobi condition and the existence of a solution to a certain Riccati equation.
TL;DR: In this article, it is shown how to steer the state of a system from a given initial state to a desired final state by observing the output of the system over a given interval.
Abstract: The well-known local results on output tracking makes it possible to control the state so that the output follows some desired path $y_d$ over some time interval $[t_0 ,t_0 + \varepsilon )$. These results have been extended to give global tracking, so that the output follows $y_d$ over a given interval $[t_0 ,t_1 ]$, for a class of systems that are sufficiently “observable.” An example is presented to illustrate how these results can be used to steer the state of the system from a given initial state to a desired final state.