H. Maurer

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All published works (16)

Optimal control of tumour-immune model with time-delay and immuno-chemotherapy

2019-02-21

Fathalla A. Rihan , S. Lakshmanan , H. Maurer

Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment

2010-05-30

Urszula Ledzewicz , J. M. Marriott , H. Maurer +1 more

Journal Article Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment Get access Urszula Ledzewicz, Urszula Ledzewicz † Department of Mathematics and Statistics, Southern Illinois University … Journal Article Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment Get access Urszula Ledzewicz, Urszula Ledzewicz † Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA †Corresponding author. Email: [email protected] Search for other works by this author on: Oxford Academic Google Scholar John Marriott, John Marriott Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA Search for other works by this author on: Oxford Academic Google Scholar Helmut Maurer, Helmut Maurer Department of Mathematics and Statistics, Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms Universität Münster, Einsteinstrasse 62, D-48149 Münster, Germany Search for other works by this author on: Oxford Academic Google Scholar Heinz Schättler Heinz Schättler Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130-4899, USA Search for other works by this author on: Oxford Academic Google Scholar Mathematical Medicine and Biology: A Journal of the IMA, Volume 27, Issue 2, June 2010, Pages 157–179, https://doi.org/10.1093/imammb/dqp012 Published: 01 June 2010 Article history Received: 23 October 2008 Revision received: 06 February 2009 Accepted: 27 April 2009 Published: 01 June 2010

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

2008-03-19

Laurenz Göllmann , Daniela Kern , H. Maurer

Abstract Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum … Abstract Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large‐scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.

Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems

2005-10-05

H. Maurer , Jochem Zowe

External optimal control of self-organisation dynamics in a chemotaxis reaction diffusion system

2004-12-01

Dirk Lebiedz , H. Maurer

Detailed quantitative understanding and specific external control of cellular behaviour are general long-term goals of modem bioscience research activities in systems biology. Pattern formation and self-organisation processes both in single … Detailed quantitative understanding and specific external control of cellular behaviour are general long-term goals of modem bioscience research activities in systems biology. Pattern formation and self-organisation processes both in single cells and in distributed cell populations are phenomena which are highly significant for the functionality of life, because life requires to maintain a highly organised spatiotemporal system structure. In particular chemotaxis is crucial for various biological aspects of intercellular signalling and cell aggregation. As an example for model based control of self-organising biological systems, we describe numerical optimal control of E. coli bacterial chemotaxis based on a 1-D two-component partial differential equation (PDE) model of reaction diffusion type. We present a numerical scheme to force cell aggregation patterns to particular desired results by applying a boundary influx control of chemoattractant without interfering with the system itself. Optimal controls are numerically computed by using a specially tailored interior point optimisation technique applied to a direct collocation discretisation of the control function and the PDE constraint. The objective to be minimised is the deviation of a desired cell distribution from the cell density, which results from the dynamics of the controlled system.

Second-Order Sufficient Conditions for State-Constrained Optimal Control Problems

2004-11-05

Kazimierz Malanowski , H. Maurer , Sabine Pickenhain

Second order sufficient conditions and sensitivity analysis for optimal multiprocess control problems

2000-01-01

Dirk Augustin , H. Maurer

Second order sufficient optimality conditions (SSC) are derived for optima! multiprocess eontroi problems. For that pur pose the multiprocess control problem is transformed into a single stage control problem with … Second order sufficient optimality conditions (SSC) are derived for optima! multiprocess eontroi problems. For that pur pose the multiprocess control problem is transformed into a single stage control problem with augmented state variables which com prise the state variables of all individua.l sta.ges as well as the switch ing times as choice variables. This tra.nsforma.tion allows to a.pply the known SSC for single sta.ge control problems. A numerical t est of SSC involves the solution of an associa.ted R.iccati equation together with boundary conditions adapted to the multiprocess . Sensitivity analysis of pa.ra.metric multiprocess problems can be based on SSC. A numerical example of the optima! two- stage control of a robot illustrates both SSC and sensitivity analysis.

Solution differentiability for parametric nonlinear control problems with control-state constraints

1995-08-01

H. Maurer , Hans Josef Pesch

The non‐linear beam via optimal control with bounded state variables

1991-01-01

H. Maurer , Hans D. Mittelmann

Abstract The non‐linear beam with bounded deflection is considered as an optimal control problem with bounded state variables. The theory of necessary optimality conditions leads to boundary value problems with … Abstract The non‐linear beam with bounded deflection is considered as an optimal control problem with bounded state variables. The theory of necessary optimality conditions leads to boundary value problems with jump conditions which are solved by multiple‐shooting techniques. A branching analysis is performed which exhibits the different solution structures. In particular, the second bifurcation point is determined numerically. The bifurcation diagram reveals a hysteresis‐like behaviour and explains the jumping to a different state at this bifurcation point.

Differential stability in infinite-dimensional nonlinear programming

1980-03-01

Frank Lempio , H. Maurer

First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems

1979-12-01

H. Maurer , Jochem Zowe

Optimality Conditions for the Programming Problem in Infinite Dimensions

1978-01-01

Jochem Zowe , H. Maurer

Differentiable Perturbations of Infinite Optimization Problems

1978-01-01

Frank Lempio , H. Maurer

On Optimal Control Problems with Bounded State Variables and Control Appearing Linearly

1977-05-01

H. Maurer

Necessary conditions for the switching function, holding at junction points of optimal interior and boundary arcs or at contact points with the boundary, are given. These conditions are used to … Necessary conditions for the switching function, holding at junction points of optimal interior and boundary arcs or at contact points with the boundary, are given. These conditions are used to derive necessary conditions for the optimality of junctions between interior and boundary arcs. The junction theorems obtained are similar to those developed for singular control problems in [1] and establish a duality between singular control problems and control problems with bounded state variables and control appearing linearly. The transition from unconstrained to constrained extremals is discussed with respect to the order p of the state constraint. A numerical example is given where the adjoins variables are not unique but form a convex set which is determined numerically.

Numerical solution of singular control problems using multiple shooting techniques

1976-02-01

H. Maurer

Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables

1975-06-01

H. Maurer , W. Gillessen

Common Coauthors

Coauthor	Papers Together
Jochem Zowe	3
Frank Lempio	2
Kazimierz Malanowski	1
S. Lakshmanan	1
Daniela Kern	1
Hans D. Mittelmann	1
W. Gillessen	1
Heinz Schättler	1
Fathalla A. Rihan	1
Laurenz Göllmann	1
Sabine Pickenhain	1
Urszula Ledzewicz	1
J. M. Marriott	1
Dirk Lebiedz	1
Hans Josef Pesch	1
Dirk Augustin	1

Commonly Cited References

Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems

1976-09-01

Stephen M. Robinson

This paper develops a condition for stability of the solution set of a system of nonlinear inequalities over a closed convex set in a Banach space, when the functions defining … This paper develops a condition for stability of the solution set of a system of nonlinear inequalities over a closed convex set in a Banach space, when the functions defining the inequalities are subjected to small perturbations. The condition involves the linearization of the system about a point; it is shown to be sufficient and, under a weak additional hypothesis, also necessary for stability. Quantitative estimates for the changes in the solution set are obtained.

Nonlinear Programming--Sequential Unconstrained Minimization Techniques

1969-08-01

M. J. D. Powell

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First Order Conditions for General Nonlinear Optimization

1976-06-01

Stephen M. Robinson

We show how first order optimality conditions for a very general nonlinear optimization problem may be derived in a conceptually simple and unified manner in terms of certain multivalued functions … We show how first order optimality conditions for a very general nonlinear optimization problem may be derived in a conceptually simple and unified manner in terms of certain multivalued functions associated with the problem. Necessary conditions for general problems and sufficient conditions for convex problems are developed, and the classical multiplier conditions are shown to be related in a simple way to these.

Bemerkungen zur Lagrangeschen Funktionaldifferentialgleichung

1974-01-01

Frank Lempio

Generalized Kuhn–Tucker Conditions for Mathematical Programming Problems in a Banach Space

1969-05-01

Monique Guignard

Generalized Kuhn–Tucker conditions stated in this paper correspond to the optimality conditions for mathematical programming problems in a Banach space. Constraint qualifications given before can be regarded as special cases … Generalized Kuhn–Tucker conditions stated in this paper correspond to the optimality conditions for mathematical programming problems in a Banach space. Constraint qualifications given before can be regarded as special cases of the present constraint qualification introduced to prove the necessity. Pseudoconvexity of the constraint set rather than convexity is required for sufficiency. In case this hypothesis fails to be satisfied, second order optimality conditions are sufficient for an isolated local optimum.

New necessary conditions of optimality for control problems with state-variable inequality constraints

1971-08-01

D. Jacobson , Milind M. Lele , Jason L. Speyer

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Regularity and Stability for Convex Multivalued Functions

1976-05-01

Stephen M. Robinson

Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex … Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions.

Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables

1975-06-01

H. Maurer , W. Gillessen

Differential Stability in Nonlinear Programming

1977-02-01

Jacques Gauvin , Jon W. Tolle

This paper consists of a study of stability and differential stability in nonconvex programming. For a program with equality and inequality constraints, upper and lower bounds are estimated for the … This paper consists of a study of stability and differential stability in nonconvex programming. For a program with equality and inequality constraints, upper and lower bounds are estimated for the potential directional derivatives of the perturbation function (or the extremal-value function). These results are obtained' with the help of a constraint qualification which is shown to be necessary and sufficient to have bounded multipliers. New results on the continuity of the perturbation function are also obtained.

First and second order sufficient optimality conditions in mathematical programming and optimal control

1981-01-01

Helmut Maurer

Sensitivity analysis for parametric control problems with control-state constraints

1996-05-01

Kazimierz Malanowski , Helmut Maurer

On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming

2005-04-28

Andreas Wächter , Lorenz T. Biegler

Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems

2005-10-05

H. Maurer , Jochem Zowe

Nonlinear programming applied to state-constrained optimization problems

1973-07-01

D. O. Norris

Stability Theory for Systems of Inequalities. Part I: Linear Systems

1975-10-01

Stephen M. Robinson

This paper deals with the stability of systems of linear inequalities in partially ordered Banach spaces when the data are subjected to small perturbations. We show that a certain condition … This paper deals with the stability of systems of linear inequalities in partially ordered Banach spaces when the data are subjected to small perturbations. We show that a certain condition is necessary and sufficient for such stability. For some of the more important special cases, this condition is computationally verifiable; it reduces to the classical full-row-rank condition in the case of equations alone. In addition, we give quantitative estimates for the magnitudes of the changes in the solution sets in terms of the magnitudes of the perturbations.

Regularity and stability for the mathematical programming problem in Banach spaces

1979-03-01

Jochem Zowe , S. Kurcyusz

Numerical solution of singular control problems using multiple shooting techniques

1976-02-01

H. Maurer

Sensitivity analysis of solutions to optimization problems in Hilbert spaces with applications to optimal control and estimation

1992-09-01

Kazufumi Ito , Karl Kunisch

A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis

2014-02-15

Fathalla A. Rihan , D.H. Abdel Rahman , S. Lakshmanan +1 more

New Jacobi-type necessary and sufficient conditions for singular optimization problems

1970-08-01

J. P. Mcdanell , William F. Powers

Jacobi-type conditions for singular optimization problems obtained by transformation to nonsingular form adaptable to linear quadratic optimal control theory Jacobi-type conditions for singular optimization problems obtained by transformation to nonsingular form adaptable to linear quadratic optimal control theory

Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients

1991-01-01

Ralph Ford , Douglas A. Lauffenburger

Quantitative Analysis of Bacterial Migration in Chemotaxis

1972-03-01

Frederick W. Dahlquist , Peter S. Lovely , Daniel E. Koshland

Complex patterns formed by motile cells of Escherichia coli

1991-02-01

Elena O. Budrene , Howard C. Berg

OPTIMAL CONTROL OF MIXED IMMUNOTHERAPY AND CHEMOTHERAPY OF TUMORS

2008-03-01

L. G. de Pillis , K. Renee Fister , Wenling Gu +5 more

We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of … We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of an optimal control for this model and provide necessary conditions for the optimal control triple for simultaneous application of chemotherapy, tumor infiltrating lymphocyte (TIL) therapy, and interleukin-2 (IL-2) treatment. We discuss numerical results for the combination of the chemo-immunotherapy regimens. We find that the qualitative nature of our results indicates that chemotherapy is the dominant intervention with TIL interacting in a complementary fashion with the chemotherapy. However, within the optimal control context, the interleukin-2 treatment does not become activated for the estimated parameter ranges.

Optimal control of time‐delay systems by dynamic programming

1992-01-01

S.A. Dadebo , Rein Luus

Abstract The use of iterative dynamic programming employing systematic region contraction and accessible grid points is investigated for the optimal control of time‐delay systems. At the time of generating the … Abstract The use of iterative dynamic programming employing systematic region contraction and accessible grid points is investigated for the optimal control of time‐delay systems. At the time of generating the grid points for the state variables, the corresponding delayed variables at each time stage are also generated and stored in memory. Then, when applying dynamic programming, a linear approximation is used to obtain the initial profile for the delayed variables during integration. This procedure was tested with four problems of different complexity. In each case the optimal control policy is easily obtained and the results compare very favourably with those reported in the literature using other computational procedures.

A generalized gradient method for optimal control problems with inequality constraints and singular arcs

1972-02-01

R.К. Mehra , Robert H. Davis

The steepest descent methods of Bryson and Ho [1] and Kelly [6] and the conjugate gradient method of Lasdon, Mitter, and Waren [3] use control variables as the independent variables … The steepest descent methods of Bryson and Ho [1] and Kelly [6] and the conjugate gradient method of Lasdon, Mitter, and Waren [3] use control variables as the independent variables in the search procedure. The inequality constraints are often handled via penalty functions which result in poor convergence. Special difficulties are encountered in handling state variable inequality constraints and singular arcs [1]. This paper shows that these difficulties arise due to the exclusive use of control variables as the independent variables in the search procedure. An algorithm based on the generalized reduced gradient (GRG) algorithm of Abadie and Carpentier [5] and Abadie [7] for nonlinear programming is proposed to solve these problems. The choice of the independent variables in this algorithm is dictated by the constraints on the problem and could result in different combinations of state and control variables as independent variables along different parts of the trajectory. The gradient of the cost function with respect to the independent variables, called the generalized gradient, is calculated by solving a set of equations similar to the Euler-Lagrange equations. The directions of search are determined using gradient projection and the conjugate gradient method. Two numerical examples involving state variable inequality constraints are solved [2]. The method is then applied to two examples containing singular arcs and it is shown that these problems can be handled as regular problems by choosing some of the state variables as the independent variables. The relationship of the method to the reduced gradient method of Wolfe [4] and the generalized reduced method of Abadie [7] for nonlinear programming is shown.

Macrophage T lymphocyte interactions in the anti-tumor immune response: a mathematical model.

1985-04-01

Rob J. de Boer , Paulien Hogeweg , Hub F. J. Dullens +2 more

Abstract In this paper we present a model of the macrophage T lymphocyte interactions that generate an anti-tumor immune response. The model specifies i) induction of cytotoxic T lymphocytes, ii) … Abstract In this paper we present a model of the macrophage T lymphocyte interactions that generate an anti-tumor immune response. The model specifies i) induction of cytotoxic T lymphocytes, ii) antigen presentation by macrophages, which leads to iii) activation of helper T cells, and iv) production of lymphoid factors, which induce a) cytotoxic macrophages, b) T lymphocyte proliferation, and c) an inflammation reaction. Tumor escape mechanisms (suppression, antigenic heterogeneity) have been deliberately omitted from the model. This research combines hitherto unrelated or even contradictory data within the range of behavior of one model. In the model behavior, helper T cells play a crucial role: Tumors that differ minimally in antigenicity (i.e., helper reactivity) can differ markedly in rejectability. Immunization yields protection against tumor doses that would otherwise be lethal, because it increases the number of helper T cells. The magnitude of the cytotoxic effector cell response depends on the time at which helper T cells become activated: early helper activity steeply increases the magnitude of the immune response. The type of cytotoxic effector cells that eradicates the tumor depends on tumor antigenicity: lowly antigenic tumors are attacked mainly by macrophages, whereas large highly antigenic tumors can be eradicated by cytotoxic T lymphocytes only.

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Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models

2009-01-01

Urszula Ledzewicz , James Munden , Heinz Schättler

The problem of scheduling a given amount of angiogenic inhibitors is consideredas an optimal control problem with the objective of maximizing the achievable tumor reduction.For a dynamical model for the … The problem of scheduling a given amount of angiogenic inhibitors is consideredas an optimal control problem with the objective of maximizing the achievable tumor reduction.For a dynamical model for the evolution of the carrying capacity of the vasculature formulatedin [15] optimal controls are computed for both a Gompertzian and logistic model of tumor growth.While optimal controls for the Gompertzian model typically contain a segment along whichthe control is singular, for the logistic model optimal controls are bang-bang with atmost two switchings.

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Numerical computation of singular control problems with application to optimal heating and cooling by solar energy

1979-03-01

Hans Joachim Oberle

AntiAngiogenic Therapy in Cancer Treatment as an Optimal Control Problem

2007-01-01

Urszula Ledzewicz , Heinz Schättler

Antiangiogenic therapy is a novel treatment approach in cancer therapy that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In this … Antiangiogenic therapy is a novel treatment approach in cancer therapy that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In this paper a mathematical model for antiangiogenic treatments based on a biologically validated model by Hahnfeldt et al. is analyzed as an optimal control problem and a full solution of the problem is given. Geometric methods from optimal control theory are utilized to arrive at the solution.

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Numerical Methods for Two-Point Boundary Value Problems

2019-08-29

Herbert B. Keller

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Controllability and the Singular Problem

1964-01-01

Henry Hermes

Previous article Next article Controllability and the Singular ProblemH. HermesH. Hermeshttps://doi.org/10.1137/0302022PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. … Previous article Next article Controllability and the Singular ProblemH. HermesH. Hermeshttps://doi.org/10.1137/0302022PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana (2), 5 (1960), 102–119 MR0127472 0112.06303 Google Scholar[2] H. Hermes and , G. Haynes, On the nonlinear control problem with control appearing linearly, J. Soc. Indust. Appl. Math. Ser. A Control, 1 (1963), 85–108 (1963) MR0170738 0145.12602 LinkGoogle Scholar[3] C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann., 67 (1909), 355–386 MR1511534 CrossrefGoogle Scholar[4] W. L. Chow, Über Systeme von linear partiellen Differentialgleiehungen erster Ordnung, Math. Ann., (1940), 95–105 Google Scholar[5] J. P. LaSalle, The time optimal control problemContributions to the theory of nonlinear oscillations, Vol. V, Princeton Univ. Press, Princeton, N.J., 1960, 1–24 MR0145169 Google Scholar[6] R. E. Kalman, , Y. C. Ho and , K. S. Narendra, Controllability of linear dynamical systems, Contributions to Differential Equations, 1 (1963), 189–213 MR0155070 0151.13303 Google Scholar[7] Earl A. Coddington and , Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955xii+429 MR0069338 0064.33002 Google Scholar[8] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963vi+153 MR0163331 0108.10401 CrossrefGoogle Scholar[9] E. B. Lee and , L. Markus, Optimal control for nonlinear processes, Arch. Rational Mech. Anal., 8 (1961), 36–58 MR0128571 0099.08703 CrossrefISIGoogle Scholar[10] R. E. Kalman, Discussion of Optimal control for nonlinear processes, Trans. ASME Ser. D. J. Basic Engrg., 84 (1962), Google Scholar[11] L. S. Pontryagin, , V. G. Boltyanskii, , R. V. Gamkrelidze and , E. F. Mishchenko, The mathematical theory of optimal processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962viii+360 MR0166037 0102.32001 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Controlling Nonlinear Infinite-Dimensional Systems via the Initial StateNeil D. 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Numerical solution of minimax optimal control problems by multiple shooting technique

1986-08-01

Hans Joachim Oberle

Real‐time computation of feedback controls for constrained optimal control problems. part 1: Neighbouring extremals

1989-04-01

Hans Josef Pesch

Abstract A numerical method is developed for the real‐time computation of neighbouring optimal feedback controls for constrained optimal control problems. The first part of this paper presents the theory of … Abstract A numerical method is developed for the real‐time computation of neighbouring optimal feedback controls for constrained optimal control problems. The first part of this paper presents the theory of neighbouring extremals. Besides a survey of the theory of neighbouring extremals, special emphasis is laid on the inclusion of complex constraints, e.g. state and control variable inequality constraints and discontinuities of the system equations at interior points. The numerical treatment of these constraints is particularly emphasized. The linearization of all necessary conditions of optimal control theory leads to a linear, mulitpoint, boundary value problem with linear jump conditions that is especially well suited for numerical treatment.

Solution Differentiability for Nonlinear Parametric Control Problems

1994-11-01

Helmut Maurer , Hans Josef Pesch

Perturbed nonlinear control problems with data depending on a vector parameter are considered. Using second-order sufficient optimality conditions, it is shown that the optimal solution and the adjoint multipliers are … Perturbed nonlinear control problems with data depending on a vector parameter are considered. Using second-order sufficient optimality conditions, it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof exploits the close connections between solutions of a Riccati differential equation and shooting methods for solving the associated boundary value problem. Solution differentiability provides a firm theoretical basis for numerical feedback schemes that have been developed for computing neighbouring extremals. The results are illustrated by an example that admits two extremal solutions. Second-order sufficient conditions single out one optimal solution for which a sensitivity analysis is carried out.

Two-Dimensional Chemotherapy Simulations Demonstrate Fundamental Transport and Tumor Response Limitations Involving Nanoparticles

2004-11-18

John P. Sinek , Hermann B. Frieboes , Xiaoming Zheng +1 more

Modified quasilinearization and optimal initial choice of the multipliers part 2?Optimal control problems

1970-11-01

A. Miele , Ramaswamy R. Iyer , K. H. Well

Dynamics Analysis and Limit Cycle in a Delayed Model for Tumor Growth with Quiescence

2006-02-27

Radouane Yafia

Realistic modelled optimal control problems in aerospace engineering - a challenge to the necessary optimality

1996-01-01

Kurt Chudej

A control problem for a hypersonic space vehicle is used to illustrate the need for a generalization of the necessary optimality conditions in the accurate numerical solution of more realistic … A control problem for a hypersonic space vehicle is used to illustrate the need for a generalization of the necessary optimality conditions in the accurate numerical solution of more realistic models for optimal control problems in aerospace engineering.

On the optimal control of systems having pure time delays and singular arcsI, Some necessary conditions for optimality†

1972-11-01

M. A. SOUMAN , W. Harmon Ray

Abstract The problem of singular arcs in the optimal control of systems having time delays is discussed. An expanded Legendre-Clebsch necessary condition is derived, and junction conditions for the boundary … Abstract The problem of singular arcs in the optimal control of systems having time delays is discussed. An expanded Legendre-Clebsch necessary condition is derived, and junction conditions for the boundary between non-singular and singular arcs is presented for a general class of problems. These and other ideas are illustrated by a number of examples.

Manipulation of Self-Aggregation Patterns and Waves in a Reaction-Diffusion System by Optimal Boundary Control Strategies

2003-11-10

Dirk Lebiedz , Ulrich Brandt-Pollmann

Reaction-diffusion systems are of considerable importance in many areas of physical sciences. For many reasons, an external manipulation of the dynamics of these processes is desirable. Here we show numerically … Reaction-diffusion systems are of considerable importance in many areas of physical sciences. For many reasons, an external manipulation of the dynamics of these processes is desirable. Here we show numerically how spatiotemporal behavior like pattern formation and wave propagation in a two component nonlinear reaction-diffusion model of bacterial chemotaxis can be externally controlled. We formulate the control goal as an objective functional and apply numerical optimization for the solution of the resulting control problem.

Nonlinear Programming in Banach Space

1967-03-01

Pravin Varaiya

Previous article Next article Nonlinear Programming in Banach SpaceP. P. VaraiyaP. P. Varaiyahttps://doi.org/10.1137/0115028PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. W. Kuhn and , A. W. Tucker, Nonlinear programming, Proceedings … Previous article Next article Nonlinear Programming in Banach SpaceP. P. VaraiyaP. P. Varaiyahttps://doi.org/10.1137/0115028PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. W. Kuhn and , A. W. Tucker, Nonlinear programming, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, 481–492, vol. 5 MR0047303 0044.05903 Google Scholar[2] Kenneth J. Arrow, , Leonid Hurwicz and , Hirofumi Uzawa, Studies in linear and non-linear programming, With contributions by H. B. Chenery, S. M. Johnson, S. Karlin, T. Marschak, R. M. Solow. Stanford Mathematical Studies in the Social Sciences, vol. II, Stanford University Press, Stanford, Calif., 1958vii+229 MR0108399 0091.16002 Google Scholar[3] Kenneth J. Arrow and , Leonid Hurwicz, Reduction of constrained maxima to saddle-point problems, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. V, University of California Press, Berkeley and Los Angeles, 1956, 1–20 MR0084938 0070.05804 Google Scholar[4] N. Dunford and , J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1964 Google Scholar[5] J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York, 1960xiv+361 MR0120319 0100.04201 Google Scholar[6] Kenneth J. Arrow, , Leonid Hurwicz and , Hirofumi Uzawa, Constraint qualifications in masimization problems, Naval Res. Logist. Quart., 8 (1961), 175–191 MR0129481 0129.34103 CrossrefGoogle Scholar[7] P. Varaiya, Nonlinear programming and optimal control, ERL Tech. Memo., M-129, University of California, Berkeley, 1965 Google Scholar[8] P. Varaiya, An extremal problem in Banach space with applications to optimal control, ERL Tech. Memo., M-180, University of California, Berkeley, 1966 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Closed-Form Expressions for Projectors onto Polyhedral Sets in Hilbert SpacesKrzysztof E. 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Model for chemotaxis

1971-02-01

Evelyn Fox Keller , Lee A. Segel

Determination of the inertia of a partitioned Hermitian matrix

1968-01-01

Emilie V. Haynsworth

State Constraints in Optimal Control: A Case Study in Proximal Normal Analysis

1987-11-01

Frank H. Clarke , Philip D. Loewen

We consider an optimal control problem on a given interval $[0,T]$ whose trajectories must satisfy the state constraint $g(t,x(t)) \leqq 0$ a.e. Infinite-dimensional perturbations of this constraint give rise to … We consider an optimal control problem on a given interval $[0,T]$ whose trajectories must satisfy the state constraint $g(t,x(t)) \leqq 0$ a.e. Infinite-dimensional perturbations of this constraint give rise to a value function V, whose epigraph is a closed set containing sensitivity information, controllability and penalization results and even necessary conditions for optimality. Studying epi V when the domain of V is $L^2 [0,T]$ and $AC^2 [0,T]$ allows the derivation of a variety of necessary conditions, each with its own merits, and provides a concrete illustration of the intricacies of infinite-dimensional proximal normal analysis.

The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation

2006-03-28

Alberto d’Onofrio , Alberto Gandolfi

Sufficient Conditions for Nonnegativity of the Second Variation in Singular and Nonsingular Control Problems

1970-08-01

D. Jacobson

Sufficient conditions for nonnegativity of the second variation in singular and nonsingular control problems are presented; these conditions are in the form of equalities and differential inequalities. Control problem examples … Sufficient conditions for nonnegativity of the second variation in singular and nonsingular control problems are presented; these conditions are in the form of equalities and differential inequalities. Control problem examples illustrate the use of the new conditions. The relationships of the new conditions to existing necessary conditions of optimality for singular and nonsingular problems are discussed. When applied to nonsingular control problems, it is shown that the conditions are sufficient to ensure the boundedness of the solution of the well-known matrix Riccati differential equation.

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Runge-Kutta methods in optimal control and the transformed adjoint system

2000-12-01

William W. Hager

A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting

1974-08-01

Peter Deuflhard

Optimality, stability, and convergence in nonlinear control

1995-05-01

Asen L. Dontchev , William W. Hager , Aubrey B. Poore +1 more