Showing papers in "Applied Mathematics and Optimization in 1980"
TL;DR: In this article, a semigroup formulation of parabolic boundary value problems is presented, which allows to translate known regularity results of the elliptic theory directly into regularity result for the parabolic solutions.
Abstract: This paper presents and abstract semigroup formulation ofparabolic boundary value problems. Smoothness of solutions, represented by a semigroup formula, is the primary object of discussion. The generality of our approach enables us to treat in a unified manner theregularity of solutions to parabolic equations for a large variety of nonhomogeneous boundary value problems. In particular, the approach presented here allows us to translate known regularity results of the elliptic theory directly into regularity results for the parabolic solutions. On the one hand, our theory recaptures known regularity results of the parabolic solutions over smooth spatial domains. On the other hand, however, our theory also covers the case of conical spatial domains, for which the standard assumption ofC∞-boundaries is violated by suitable application of recent relevant results of elliptic theory for such domains. In the concluding section, an application of our general theory to a boundary control problem with a quadratic performance index is presented.
174 citations
TL;DR: In this article, a parabolic equation defined on a bounded domain is considered, with input acting on the boundary expressed as a specified feedback of the solution, and both Dirichlet and mixed (in particular, Neumann) boundary conditions are treated.
Abstract: A parabolic equation defined on a bounded domain is considered, with input acting on theboundary expressed as a specifiedfeedback of the solution. Both Dirichlet and mixed (in particular, Neumann) boundary conditions are treated. Algebraic conditions based on the finitely many unstable eigenvalues are given, ensuring the existence ofboundary vectors, for which all the solutions to theboundary feedback parabolic equation decay exponentially to zero ast→+∞ in (essentially) the strongest possible space norm. A semigroup approach is employed.
137 citations
TL;DR: In this article, a cutting plane algorithm is developed for reverse convex programs with disconnected feasible regions and basic solutions are defined and properties of the latter and of the convex hull of the feasible region are derived.
Abstract: Reverse convex programs generally have disconnected feasible regions. Basic solutions are defined and properties of the latter and of the convex hull of the feasible region are derived. Solution procedures are discussed and a cutting plane algorithm is developed.
99 citations
TL;DR: In this article, the stability properties of the extremal value function are studied for infinite-dimensional nonlinear optimization problems with differentiable perturbations in the objective function and in the constraints.
Abstract: In this paper stability properties of the extremal value function are studied for infinite-dimensional nonlinear optimization problems with differentiable perturbations in the objective function and in the constraints In particular, upper and lower bounds for the directional derivative of the extremal value function as well as necessary and sufficient conditions for the existence of the directional derivative are given
76 citations
TL;DR: In this paper, it was shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytoope defined by only the linear constraints.
Abstract: A constraintg(x)⩾0 is said to be a reverse convex constraint if the functiong is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected. It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytope defined by only the linear constraints. The only possible edges which can contain such an optimal solution are characterized in relation to the best feasible vertex of the polytope defined by only the linear constraints. This characterization then provides a finite algorithm for finding a globally optimal solution.
62 citations
TL;DR: In this article, it was shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemsxk+1=Ax k.
Abstract: This paper is devoted to a study of the properties of the equationA*FA−F=−G, where F∈L(ℌ) is unknown, A∈L(ℌ), G∈L(ℌ) is positive andℌ is a Hilbert space. It is shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemxk+1=Ax k . The relationships between stability of such a system and stability of a continuous-time system generated by a strongly continuous semigroup are given also. As an example the case of the delayed system in Rn\(R^n \dot x\left( t \right) = A_0 x\left( t \right) + A_1 x\left( {t - 1} \right)\) is considered.
38 citations
TL;DR: In this article, the authors consider a constrained stochastic control problem (stochastic bang-bang control), which has been studied recently by many authors, including Ruzicka, Zwonkin-Krylov, and Ikeda-Watanabe.
Abstract: We consider a constrained stochastic control problem (stochastic bang-bang control) studied recently by many authors Ruzicka [1], Zwonkin-Krylov [2], and Ikeda-Watanabe [3], among others. Our techniques are quite different from theirs, and we obtain in addition some results complementing theirs which would appear to be of independent interest as well.
35 citations
TL;DR: Finite-element approximation of a Dirichlet type boundary control problem for parabolic systems is considered and it turns out that these estimates are optimal with respect to the approximation theoretic properties.
Abstract: Finite-element approximation of a Dirichlet type boundary control problem for parabolic systems is considered. An approach based on the direct approximation of an input-output semigroup formula is applied. Error estimates inL2[OT; L2(Ω)] andL2[OT; L2(Γ)] norms are derived for optimal state and optimal control, respectively. It turns out that these estimates areoptimal with respect to the approximation theoretic properties.
31 citations
TL;DR: In this paper, the authors considered quadratic control problems for parabolic equations with state constraints and showed that the optimal control is continuous in time with the values inL2(Ω) and its time derivative belongs toL2[OT×Ω.
Abstract: Quadratic control problems for parabolic equations withstate constraints are considered. Regularity (smoothness) of the optimal solution is investigated. It is shown that the optimal control is continuous in time with the values inL2(Ω) and its time derivative belongs toL2[OT×Ω].
31 citations
TL;DR: In this article, the authors considered the optimal control of continuous-time Markov processes and established that the minimum expected cost for the separated problem equals the infimum of expected costs for the original problem with partially observed states.
Abstract: This paper is concerned with the optimal control of continuous-time Markov processes. The admissible control laws are based on white-noise corrupted observations of a function on the state processes. A “separated” control problem is introduced, whose states are probability measures on the original state space. The original and separated control problems are related via the nonlinear filter equation. The existence of a minimum for the separated problem is established. Under more restrictive assumptions it is shown that the minimum expected cost for the separated problem equals the infimum of expected costs for the original problem with partially observed states.
29 citations
TL;DR: In this paper, a class of implementable penalty function and multiplier methods for nonconvex nonlinear programming problems is presented, which, under suitable assumptions, produce a sequence of points converging to a strong local minimum, regardless of the location of the initial guess.
Abstract: This paper deals with penalty function and multiplier methods for the solution of constrained nonconvex nonlinear programming problems. Starting from an idea introduced several years ago by Polak, we develop a class of implementable methods which, under suitable assumptions, produce a sequence of points converging to a strong local minimum for the problem, regardless of the location of the initial guess. In addition, for sequential minimization type multiplier methods, we make use of a rate of convergence result due to Bertsekas and Polyak, to develop a test for limiting the growth of the penalty parameter and thereby prevent ill-conditioning in the resulting sequence of unconstrained optimization problems.
TL;DR: In this paper, the authors characterize the situation wherein a subspaceS of a separable Hilbert state space is holdable under the abstract linear autonomous control system, where A is the infinitesimal generator of a C0-semigroup of operators and B is a bounded linear operator mapping a Hilbert space Ω into X.
Abstract: In this paper we characterize the situation wherein a subspaceS of a separable Hilbert state space is holdable under the abstract linear autonomous control system\(\dot x = Ax + Bu\), whereA is the infinitesimal generator of aC0-semigroup of operators and whereB is a bounded linear operator mapping a Hilbert space Ω intoX. WhenS⊥∩D(A*) is dense inS⊥, it is shown that a necessary (but insufficient) condition for holdability is (1):\(A[S \cap D\left( A \right)] \subset \bar S + B\Omega\). A stronger condition than (1) is shown to be sufficient for a type of approximate holdability. In the finite dimensional setting, (1) reduces to (A, B)-invariance, which is known to be equivalent to the existence of a (bounded) linear feedback control law which achieves holdability inS. We prove that this equivalence holds in infinite dimensions as well, whenA is bounded and the linear spacesS, BΩ andS+ BΩ are closed.
TL;DR: In this paper, the authors examined the agreement ratio of Borda score vectors over sets of profiles of linear orders, where the restriction toB of the ranking overA produced byw operating ons equals the restriction overB produced byv operating on the restriction ofs toB. The limiting agreement ratios that are considered for (m, k) in {(3,2,4,2),(4,3)} are uniquely maximized by pairs of Bordaa score vectors and are minimized when (w,v) is either ((1,0,⋯
Abstract: Positional score vectorsw=(w1,⋯,w m ) for anm-element setA, andv=(v1,⋯,v k ) for ak-element proper subsetB ofA, agree at a profiles of linear orders onA when the restriction toB of the ranking overA produced byw operating ons equals the ranking overB produced byv operating on the restriction ofs toB. Givenw1>wmandv1>v k , this paper examines the extent to which pairs of nonincreasing score vectors agree over sets of profiles. It focuses on agreement ratios as the number of terms in the profiles becomes infinite. The limiting agreement ratios that are considered for (m, k) in {(3,2),(4,2),(4,3)} are uniquely maximized by pairs of Borda (linear, equally-spaced) score vectors and are minimized when (w,v) is either ((1,0,⋯,0),(1,⋯,1,0)) or ((1,,⋯,1,0),(1,0,⋯,0)).
TL;DR: In this article, non-anticipative representations of Gaussian random fields equivalent to the two-parameter Wiener process are defined, and necessary and sufficient conditions for their existence derived.
Abstract: Nonanticipative representations of Gaussian random fields equivalent to the two-parameter Wiener process are defined, and necessary and sufficient conditions for their existence derived. When such representations exist they provide examples of canonical representations of multiplicity one. In contrast to the one-parameter case, examples are given where nonanticipative representations do not exist. Nonanticipative representations along increasing paths are also studied.
TL;DR: In this article, generalizations of well-known conditions for controllability of linear abstract autonomous systems defined on Banach spaces to the case where there is a closed non-invertible operator at the derivative are established.
Abstract: Generalizations of well-known conditions for controllability of linear abstract autonomous systems defined on Banach spaces to the case where there is a closed non invertible operator at the derivative are established. Presence of this operator implies that suitable controllers must be chosen. If the operators entering in the equation satisfy certain hypotheses, approximate controllability by use of this class of controllers is expressed only in terms of the coefficients of the system. In particular, approximate controllability in finite time is then equivalent to approximate controllability, according to the usual definition, of a corresponding non degenerate system. This is the case, for example, when the concerned spaces are finite dimensional. Some applications to partial differential equations are given.
TL;DR: In this paper, the Clarke generalized gradient of the marginal function of a nonconvex optimization problem with respect to usual and non-usual parameters is computed and Lagrange multipliers are involved in this formula.
Abstract: We compute the Clarke generalized gradient of the marginal function of a nonconvex optimization problem with respect to usual and non usual parameters and we show how Lagrange multipliers are involved in this formula.
TL;DR: In this article, the boundary values of the solutions of elliptic equations were reported in the course of a course at the University of California, at UCLA, and the present article is a summary of the report.
Abstract: In May 1978 Professor A. V. Balakrishnan invited me to report about boundary values of the solutions of elliptic equations in his seminar at the University of California, at UCLA. I thank him for this invitation. The present article is a summary of my report.
TL;DR: In this article, it was shown that a recent regularity theorem of Zowe/Kurcyusz cannot be sharpened by sharpening the regularity regularity conjecture.
Abstract: It is shown by example that a recent regularity theorem of Zowe/Kurcyusz cannot be sharpened.
TL;DR: In this article, the primal and dual backtrack algorithms were developed to find the order of intermodulation interference in a single-dimensional constraint integer program with variables unrestricted in sign, where coefficients of the constraint equation correspond to frequencies.
Abstract: Two or more radio signals, transmitted from a small platform (e.g., a ship, satellite, or airplane, etc.), tend to produce an intermodulation product which could distort a receive signal. The intensity of intermodulation interference due to a given intermodulation frequency is known to be closely related to the lowest order, or simply the order, of the intermodulation product which can be found by solving a single constraint integer program with variables unrestricted in sign, where coefficients of the constraint equation correspond to frequencies. Two variations of backtrack, or search enumeration, algorithms, called the primal and dual backtrack algorithms, are developed to find the order of intermodulation. Extensive computational experience with these algorithms, reflecting data from naval fleet communication scenarios and potential ramifications, are given together with another application of the dual algorithm.
TL;DR: In this article, the authors used the logarithmic transformation of Fleming [1] to discuss a specific problem of controlled diffusions, where the problem is to minimize a certain quadratic functional of the applied drift while satisfying the requirement that the place where the process exits a domain is not in a specified subset of its boundary.
Abstract: In this paper the logarithmic transformation of Fleming [1] is used to discuss a specific problem of controlled diffusions. The problem is to minimize a certain quadratic functional of the applied drift while satisfying the requirement that the place where the process exits a domain is not in a specified subset of its boundary. The main result is that the solution of this problem is given by the logarithm of a related exit probability.
TL;DR: In this paper, the authors distinguished a geometric characteristic of the unbounded domain Ω, that determines the rate of stabilization for the solution in (t>0)×Ω of the second boundary value problem for a second order parabolic equation, in which the initial function decreases sufficiently rapidly as |x|→∞.
Abstract: In this paper is distinguished a geometric characteristic of the unbounded domain Ω, that determines the rate of stabilization fort→∞ of the solution in (t>0)×Ω of the second boundary value problem for a second-order parabolic equation, in which the initial function decreases sufficiently rapidly as |x|→∞.
TL;DR: In this paper, the sensitivity of state predictions to arbitrary functional perturbations to the right-hand sides of the chosen differential equations is measured by a Riemannian measure of the distance or gap between any two states.
Abstract: When modelling any system using ordinary differential equations, the problem arises of gauging the sensitivity of state predictions to arbitrary functional perturbations to the right-hand sides of the chosen differential equations. Assuming that a suitable Riemannian measure of the distance or gap between any two states (as possible predictions) has been chosen, a scalar functionr(·) of the state is defined which characterizes the insensitivity of a nominal prediction to finite functional perturbations in the differential equations. The limiting behaviour ofr(·) near the nominal prediction defines a second rank symmetric positive definite tensorM, which provides an easily computed and convenient description of the insensitivity of the nominal predictionin each direction to ‘infinitesimal’ functional perturbations in the model. This theory is fully invariant under arbitrary smooth transformations of state variables.