Optimal Controls of 3-Dimensional Navier--Stokes Equations with State Constraints

Previous article Next article Optimal Controls for Systems with Time LagA. HalanayA. Halanayhttps://doi.org/10.1137/0306016PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. L. Kharatishvili, The maximum principle in the theory of optimal processes with time lag, Dokl. Akad. Nauk SSSR, 136 (1961), 39–42 Google Scholar[1A] 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[2] Avner Friedman, Optimal control for hereditary processes, Arch. Rational Mech. Anal., 15 (1964), 396–416 10.1007/BF00256929 MR0170744 0122.10801 CrossrefISIGoogle Scholar[3] I. A. Oziganova, On the theory of optimal control of systems with time lag, Seminar on Differential Equations with Deviating Arguments, Vol. II, University of the Friendship of Peoples, Moscow, 1963, 136–145, See also: On the theory of optimal control for problems with time lag, thesis, University of the Friendship of Peoples, Moscow, 1966. Google Scholar[4] Magnus R. Hestenes, On variational theory and optimal control theory, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 23–48 MR0184763 0151.12803 LinkGoogle Scholar[5] Rodney D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401–426 10.1007/BF00281203 MR0141863 0105.30401 CrossrefISIGoogle Scholar[6] M. Namı k Oğuztöreli, Time-lag control systems, Mathematics in Science and Engineering, Vol. 24, Academic Press, New York, 1966xii+323 MR0217394 0143.12101 Google Scholar[7] G. L. Kharatishvili, A. V. Balakrishnan and , L. W. Neustadt, A maximum principle in extremal problems with delaysMathematical Theory of Control (Proc. Conf., Los Angeles, Calif., 1967), Academic Press, New York, 1967, 26–34 MR0256240 0216.17701 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Continuity of Pontryagin Extremals with Respect to Delays in Nonlinear Optimal ControlRiccardo Bonalli, Bruno Hérissé, and Emmanuel Trélat18 April 2019 | SIAM Journal on Control and Optimization, Vol. 57, No. 2AbstractPDF (576 KB)Optimal Control Problems with Time Delays: Constancy of the HamiltonianRichard B. Vinter25 July 2019 | SIAM Journal on Control and Optimization, Vol. 57, No. 4AbstractPDF (525 KB)The Maximum Principle for Optimal Control Problems with Time DelaysA. Boccia and R. B. Vinter19 September 2017 | SIAM Journal on Control and Optimization, Vol. 55, No. 5AbstractPDF (407 KB)The Construction of the Solution of an Optimal Control Problem Described by a Volterra Integral Equation17 February 2012 | SIAM Journal on Control and Optimization, Vol. 21, No. 4AbstractPDF (1468 KB)Optimal Controls with Pseudodelays18 July 2006 | SIAM Journal on Control, Vol. 12, No. 2AbstractPDF (1227 KB)Optimal Control Problems with a System of Integral Equations and Restricted Phase Coordinates18 July 2006 | SIAM Journal on Control, Vol. 10, No. 1AbstractPDF (1661 KB)The Optimization of Trajectories of Linear Functional Differential Equations18 July 2006 | SIAM Journal on Control, Vol. 8, No. 4AbstractPDF (2777 KB) Volume 6, Issue 2| 1968SIAM Journal on Control History Submitted:06 July 1967Published online:18 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0306016Article page range:pp. 215-234ISSN (print):0036-1402Publisher:Society for Industrial and Applied Mathematics

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

Previous article Next article Time-Optimal Control of Solutions of Operational Differenital EquationsH. O. FattoriniH. O. Fattorinihttps://doi.org/10.1137/0302005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Einar Hille and , Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957xii+808 MR0089373 0392.46001 Google Scholar[2] S. Bochner and , A. E. Taylor, Linear functionals on certain spaces of abstractly valued functions, Ann. of Math. (2), 39 (1938), 913–944 MR1503445 CrossrefGoogle Scholar[3] R. Bellman, , I. Glicksberg and , O. Gross, On the “bang-bang” control problem, Quart. Appl. Math., 14 (1956), 11–18 MR0078516 0073.11501 CrossrefGoogle Scholar[4] R. S. Phillips, Perturbation theory for semi-groups of linear operators, Trans. Amer. Math. Soc., 74 (1953), 199–221 MR0054167 CrossrefISIGoogle Scholar[5] Tosio Kato, On linear differential equations in Banach spaces, Comm. Pure Appl. Math., 9 (1956), 479–486 MR0086986 0070.34602 CrossrefISIGoogle Scholar[6] 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[7] Ju. V. Egorov, Optimal control in a Banach space, Dokl. Akad. Nauk SSSR, 150 (1963), 241–244, translated in Soviet Mathematics, 4 (1963). MR0149986 0133.37103 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Equivalence between Minimal Time and Minimal Norm Control Problems for the Heat EquationShulin Qin and Gengsheng Wang15 March 2018 | SIAM Journal on Control and Optimization, Vol. 56, No. 2AbstractPDF (624 KB)Observability Inequalities from Measurable Sets for Some Abstract Evolution EquationsGengsheng Wang and Can Zhang15 June 2017 | SIAM Journal on Control and Optimization, Vol. 55, No. 3AbstractPDF (546 KB)Bang-Bang Property of Time Optimal Null Controls for Some Semilinear Heat EquationLijuan Wang and Qishu Yan3 November 2016 | SIAM Journal on Control and Optimization, Vol. 54, No. 6AbstractPDF (251 KB)Perturbations of Time Optimal Control Problems for a Class of Abstract Parabolic SystemsMarius Tucsnak, Gensgsheng Wang, and Chi-Ting Wu8 November 2016 | SIAM Journal on Control and Optimization, Vol. 54, No. 6AbstractPDF (413 KB)Attainable Subspaces and the Bang-Bang Property of Time Optimal Controls for Heat EquationsGengsheng Wang, Yashan Xu, and Yubiao Zhang4 March 2015 | SIAM Journal on Control and Optimization, Vol. 53, No. 2AbstractPDF (451 KB)Maximum Principle and Bang-Bang Property of Time Optimal Controls for Schrödinger-Type SystemsJérôme Lohéac and Marius Tucsnak28 October 2013 | SIAM Journal on Control and Optimization, Vol. 51, No. 5AbstractPDF (287 KB)$L^\infty$-Null Controllability for the Heat Equation and Its Consequences for the Time Optimal Control ProblemGengsheng Wang13 June 2008 | SIAM Journal on Control and Optimization, Vol. 47, No. 4AbstractPDF (242 KB)Characterization of Extremal Controls for Infinite Dimensional Time-Varying SystemsDiomedes Barcenas and Hugo Leiva26 July 2006 | SIAM Journal on Control and Optimization, Vol. 40, No. 2AbstractPDF (181 KB)Regularity Results for the Minimum Time Function of a Class of Semilinear Evolution Equations of Parabolic TypePaolo Albano, Piermarco Cannarsa, and Carlo Sinestrari26 July 2006 | SIAM Journal on Control and Optimization, Vol. 38, No. 3AbstractPDF (266 KB)A Remark on Existence of Solutions of Infinite-Dimensional Noncompact Optimal Control ProblemsH. O. Fattorini26 July 2006 | SIAM Journal on Control and Optimization, Vol. 35, No. 4AbstractPDF (246 KB)Nonlinear Optimal Control Problems for Parabolic EquationsAvner Friedman17 February 2012 | SIAM Journal on Control and Optimization, Vol. 22, No. 5AbstractPDF (792 KB)Nonlinear Optimal Control Problems in Heat ConductionAvner Friedman and Li-Shang Jiang17 February 2012 | SIAM Journal on Control and Optimization, Vol. 21, No. 6AbstractPDF (805 KB)Boundary Control of Parabolic Differential Equations in Arbitrary Dimensions: Supremum-Norm ProblemsKlaus Glashoff and Norbert Weck18 July 2006 | SIAM Journal on Control and Optimization, Vol. 14, No. 4AbstractPDF (1514 KB)Optimal Continuous-Parameter Stochastic ControlWendell H. Fleming18 July 2006 | SIAM Review, Vol. 11, No. 4AbstractPDF (4487 KB)Boundary Controllability of Nonlinear Hyperbolic SystemsMarco Cirinà18 July 2006 | SIAM Journal on Control, Vol. 7, No. 2AbstractPDF (1292 KB)Optimal Regulation of Linear Symmetric Hyperbolic Systems with Finite Dimensional ControlsDavid L. Russell18 July 2006 | SIAM Journal on Control, Vol. 4, No. 2AbstractPDF (1430 KB) Volume 2, Issue 1| 1964Journal of the Society for Industrial and Applied Mathematics Series A Control History Submitted:11 November 1963Published online:18 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0302005Article page range:pp. 54-59ISSN (print):0887-4603ISSN (online):2168-359XPublisher:Society for Industrial and Applied Mathematics

State Transition Tensors for Continuous-Thrust Control of Three-Body Relative Motion

Previous article Next article The Reduction of Certain Control Problems to an “Ordinary Differential” TypeJ. WargaJ. Wargahttps://doi.org/10.1137/1010036PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout Previous article Next article FiguresRelatedReferencesCited byDetails Stationarity conditions in optimal control problems class related to dynamical objects group control Cross Ref On the First- and Second-Order Conditions in Optimal Control Problems Related to Autonomous Objects Group lifecycle Cross Ref Optimal Control Problems with Time Delays: Constancy of the HamiltonianRichard B. Vinter25 July 2019 | SIAM Journal on Control and Optimization, Vol. 57, No. 4AbstractPDF (525 KB)On obtaining the stationarity conditions in an optimal control problem for a trajectory with a smooth contact with the phase boundary Cross Ref On one Method to Obtain Stationarity Conditions for an Optimal Control Problem Trajectory With a Smooth Boundary ContactIFAC-PapersOnLine, Vol. 51, No. 32 Cross Ref The Maximum Principle for Optimal Control Problems with Time DelaysA. Boccia and R. B. Vinter19 September 2017 | SIAM Journal on Control and Optimization, Vol. 55, No. 5AbstractPDF (407 KB)Problems with relaxed shifted controls and the pointwise maximum principleNonlinear Analysis: Theory, Methods & Applications, Vol. 75, No. 3 Cross Ref The Instability of Optimal Control Problems to Time DelayAlexey S. Matveev26 July 2006 | SIAM Journal on Control and Optimization, Vol. 43, No. 5AbstractPDF (340 KB)On the Characterization of Properly Relaxed Delayed ControlsJournal of Mathematical Analysis and Applications, Vol. 209, No. 1 Cross Ref Approximation of strongly relaxed minimizers with ordinary delayed controlsApplied Mathematics & Optimization, Vol. 32, No. 1 Cross Ref Jack Warga: In Appreciation14 July 2006 | SIAM Journal on Control and Optimization, Vol. 33, No. 2AbstractPDF (809 KB)Minimizing approximate original solutions for commensurate delayed controlsApplied Mathematics Letters, Vol. 7, No. 5 Cross Ref How to Properly Relax Delayed Controls Cross Ref A proper relaxation of shifted and delayed controlsJournal of Mathematical Analysis and Applications, Vol. 169, No. 2 Cross Ref Optimal controls and their discontinuities in quadratic problems of delay systemsIEEE Transactions on Automatic Control, Vol. 30, No. 7 Cross Ref Optimal Controls with PseudodelaysJ. Warga18 July 2006 | SIAM Journal on Control, Vol. 12, No. 2AbstractPDF (1227 KB)A conservative bound on the estimation error covariance matrix in the presence of correlated driving noise and correlated discrete measurement noiseJournal of Mathematical Analysis and Applications, Vol. 43, No. 2 Cross Ref References Cross Ref Volume 10, Issue 2| 1968SIAM Review History Submitted:18 October 1967Published online:18 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1010036Article page range:pp. 219-222ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

We prove an existence and uniqueness result on periodic solutions of an infinite dimensional Riccati equation.

Previous article Next article An Extended Pontryagin Principle for Control Systems whose Control Laws Contain MeasuresRaymond W. RishelRaymond W. Rishelhttps://doi.org/10.1137/0303016PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. B. Barrar, An analytic proof that Hohman-type transfer is the true minimum two impulse transfer, Astronaut. Acta, 9 (1963), 1–11 Google Scholar[2] H. O. Ladd, Jr. and , B. Friedland, Minimum fuel control of a second order linear process with a constraint on time to run, Trans. ACME Ser. D., 86 (1964), 160–168 CrossrefGoogle Scholar[3] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950xi+304 MR0033869 0040.16802 CrossrefGoogle Scholar[4] D. F. Lawden, Optimal trajectories for space navigation, Butterworths, London, 1963viii+126 MR0199011 0111.19605 Google Scholar[5] E. B. Lee, A sufficient condition in the theory of optimal control, J. Soc. Indust. Appl. Math. Ser. A Control, 1 (1963), 241–245 MR0175693 0151.13105 LinkGoogle Scholar[6] Lucien W. Neustadt, Optimization, a moment problem, and nonlinear programming, J. Soc. Indust. Appl. Math. Ser. A Control, 2 (1964), 33–53 MR0174423 0133.36701 LinkGoogle Scholar[7] 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[8] W. W. Schmaedeke, Optimal control theory for nonlinear vector differential equations with measure coefficients, Report, U-RD 63 18, Minneapolis-Honeywell, 1963 Google Scholar[9] L. Ting, Optimal orbital transfer by several impulses, Astronaut. Acta, 6 (1960), 256–265 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Nondegenerate Abnormality, Controllability, and Gap Phenomena in Optimal Control with State ConstraintsGiovanni Fusco and Monica Motta25 January 2022 | SIAM Journal on Control and Optimization, Vol. 60, No. 1AbstractPDF (673 KB)Optimal Control of Nonlinear Hybrid Systems Driven by Signed Measures with Variable Intensities and SupportsN. U. Ahmed and Shian Wang4 November 2021 | SIAM Journal on Control and Optimization, Vol. 59, No. 6AbstractPDF (726 KB)A Higher-Order Maximum Principle for Impulsive Optimal Control ProblemsM. Soledad Aronna, Monica Motta, and Franco Rampazzo12 March 2020 | SIAM Journal on Control and Optimization, Vol. 58, No. 2AbstractPDF (712 KB)Minimal Time Sequential Batch Reactors with Bounded and Impulse Controls for One or More SpeciesP. Gajardo, H. Ramírez C., and A. 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Warga18 July 2006 | Journal of the Society for Industrial and Applied Mathematics Series A Control, Vol. 3, No. 3AbstractPDF (1122 KB) Volume 3, Issue 2| 1965Journal of the Society for Industrial and Applied Mathematics Series A Control History Submitted:05 October 1964Accepted:21 December 1964Published online:18 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0303016Article page range:pp. 191-205ISSN (print):0887-4603ISSN (online):2168-359XPublisher:Society for Industrial and Applied Mathematics

A class of boundary-distributed linear control systems in Banach spaces is studied. A maximum principle for a convex control problem associated with such systems is obtained.

A neural network based optimal control synthesis is presented for solving optimal control problems with control and state constraints. The optimal control problem is transcribed into a nonlinear programming problem which is implemented with adaptive critic neural network. The proposed simulation method is illustrated by the optimal control problem of nitrogen transformation cycle model. Results show that adaptive critic based systematic approach holds promise for obtaining the optimal control with control and state constraints. INTRODUCTION Optimal control of nonlinear systems is one of the most active subjects in control theory. There is rarely an analytical solution although several numerical computation approaches have been proposed (for example, see (Polak, 1997), (Kirk, 1998)) for solving a optimal control problem. Most of the literature that deals with numerical methods for the solution of general optimal control problems focuses on the algorithms for solving discretized problems. The basic idea of these methods is to apply nonlinear programming techniques to the resulting finite dimensional optimization problem (Buskens at al., 2000). When Euler integration methods are used, the recursive structure of the resulting discrete time dynamic can be exploited in computing first-order necessary condition. In the recent years, the multi-layer feedforward neural networks have been used for obtaining numerical solutions to the optimal control problem. (Padhi at al., 2001), (Padhi et al., 2006). We have taken hyperbolic tangent sigmoid transfer function for the hidden layer and a linear transfer function for the output layer. The paper extends adaptive critic neural network architecture proposed by (Padhi at al., 2001) to the optimal control problems with control and state constraints. The paper is organized as follows. In Section 2, the optimal control problems with control and state constraints are introduced. We summarize necessary optimality conditions and give a short overview of basic result including the iterative numerical methods. Section 3 discusses discretization methods for the given optimal control problem. It also discusses a form of the resulting nonlinear programming problems. Section 4 presents a short description of adaptive critic neural network synthesis for optimal problem with state and control constraints. Section 5 consists of a nitrogen transformation model. In section 6, we apply the discussed methods to the nitrogen transformation cycle. The goal is to compare short-term and long-term strategies of assimilation of nitrogen compounds. Conclusions are presented in Section 7. OPTIMAL CONTROL PROBLEM We consider a nonlinear control problem subject to control and state constraints. Let x(t) ∈ R denote the state of a system and u(t) ∈ R the control in a given time interval [t0, tf ]. Optimal control problem is to minimize F (x, u) = g(x(tf )) + ∫ tf t0 f0(x(t), u(t))dt (1)

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

The classical Pontryagin maximum principle is a first-order necessary condition (FONC for short) for optimal controls of ordinary differential equations (ODEs for short). When the FONC degenerates, people introduce second-order necessary conditions (SONCs for short) for optimal controls. SONCs are well studied for optimal controls of systems described by ODEs and partial differential equations (PDEs for short). Some results for SONC of optimal controls for systems governed by stochastic differential equations (SDEs for short) are also obtained. However, nothing is known about the SONC for optimal controls of systems described by stochastic (infinite dimensional) evolution equations (SEEs for short). This paper aims to give a solution to this difficult problem, under some mild conditions.

We prove the existence of a value for pursuit games with state constraints. We also prove that this value is lower semicontinuous.

Contraction Mappings in the Theory Underlying Dynamic Programming

Previous article Next article An Optimal Control Problem for Systems with Differential-Difference Equation DynamicsD. W. Ross and I. Flügge-LotzD. W. Ross and I. Flügge-Lotzhttps://doi.org/10.1137/0307044PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] D. W. Ross, Masters Thesis, Optimal control of systems described by differential-difference equations, Doctoral thesis, Department of Electrical Engineering, Stanford University, Stanford, California, 1967 Google Scholar[2] R. Bellman and , J. M. Danskin, A Survey of the Mathematical Theory of Time Lag, Retarded Control, and Hereditary Processes, RAND Rep., R-256, Santa Monica, California, 1954 Google Scholar[3] N. H. Choksy, Time-lag systems—a bibliography, IRE Trans. Automatic Control, AC-5 (1960), 66–70 CrossrefGoogle Scholar[4] H. S. Tsien, Engineering Cybernetics, McGraw-Hill, New York, 1954 Google Scholar[5] Richard Bellman and , Kenneth L. Cooke, Differential-difference equations, Academic Press, New York, 1963xvi+462 MR0147745 (26:5259) 0105.06402 Google Scholar[6] M. N. Oguztöreli, A time optimal control problem for systems described by differential-difference equations, SIAM J. Control, 1 (1963), 290–310 Google Scholar[7] 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 (29:3316b) 0102.32001 Google Scholar[8] D. H. Chyung and , E. Bruce Lee, Linear optimal systems with time delays, SIAM J. Control, 4 (1966), 548–575 10.1137/0304042 MR0207425 (34:7240) 0148.33804 LinkGoogle Scholar[9] N. N. Krasovskii˘, On the analytic construction of an optimal control in a system with time lags, J. Appl. Math. Mech., 26 (1962), 50–67 MR0145171 (26:2706) CrossrefGoogle Scholar[10] N. N. 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Bernstein, Existence Theorems in Partial Differential Equations, Annals of Mathematics Studies, no. 23, Princeton University Press, Princeton, N. J., 1950ix+228 MR0037440 (12,262c) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A Convex Solution of the $H_\infty$-Optimal Controller Synthesis Problem for Multidelay SystemsMatthew M. PeetSIAM Journal on Control and Optimization, Vol. 58, No. 3 | 11 June 2020AbstractPDF (716 KB)Numerical Approximation for the Infinite-Dimensional Discrete-Time Optimal Linear-Quadratic Regulator ProblemSIAM Journal on Control and Optimization, Vol. 26, No. 2 | 14 July 2006AbstractPDF (2660 KB)A Spline Based Technique for Computing Riccati Operators and Feedback Controls in Regulator Problems for Delay EquationsSIAM Journal on Scientific and Statistical Computing, Vol. 5, No. 4 | 14 July 2006AbstractPDF (2290 KB)Linear-Quadratic Optimal Control of Hereditary Differential Systems: Infinite Dimensional Riccati Equations and Numerical ApproximationsSIAM Journal on Control and Optimization, Vol. 21, No. 1 | 17 February 2012AbstractPDF (4860 KB)Approximation Schemes for the Linear-Quadratic Optimal Control Problem Associated with Delay EquationsSIAM Journal on Control and Optimization, Vol. 20, No. 4 | 14 July 2006AbstractPDF (2889 KB)Hereditary Control Problems: Numerical Methods Based on Averaging ApproximationsSIAM Journal on Control and Optimization, Vol. 16, No. 2 | 18 July 2006AbstractPDF (3382 KB)Stability and the Infinite-Time Quadratic Cost Problem for Linear Hereditary Differential SystemsSIAM Journal on Control, Vol. 13, No. 1 | 18 July 2006AbstractPDF (2372 KB)Unconstrained Control Problems with Quadratic CostSIAM Journal on Control, Vol. 11, No. 1 | 18 July 2006AbstractPDF (1650 KB)Controllability, Observability and Optimal Feedback Control of Affine Hereditary Differential SystemsSIAM Journal on Control, Vol. 10, No. 2 | 18 July 2006AbstractPDF (2525 KB) Volume 7, Issue 4| 1969SIAM Journal on Control521-671 History Submitted:09 January 1968Accepted:09 January 1969Published online:01 August 2006 InformationCopyright © 1969 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0307044Article page range:pp. 609-623ISSN (print):0036-1402Publisher:Society for Industrial and Applied Mathematics

In this paper we analyze an optimal control problem governed by a convection diffusion equation. This problem with state constraints is discussed by adding penalty arguments involving the application of Ekeland's variational principle and finite codimensionality of certain sets. Necessary conditions for optimal control is established by the method of spike variation.