Synthesis of Optimal Control in a Mathematical Model of Economic Growth under R&D Investments

Even for suciently simple single-input control-ane systems without state constraints the subject of analytical and qualitative investigation of global optimal control synthesis is developed rather poorly in the literature on optimal control theory and its applications. The aim of this work is to present numerical-analytical and qualitative methods to construct global optimal control synthesis with reference to a nonlinear mathematical model describing interactions between the production of a company, its technology stock and R&D investments on a xed time in

[1]  H. Schättler,et al.  Geometric Optimal Control: Theory, Methods and Examples , 2012 .

[2]  A. Bratus,et al.  Optimal control synthesis in therapy of solid tumor growth , 2008 .

[3]  N. Subbotina,et al.  Estimating error of the optimal grid design in the problems of nonlinear optimal control of prescribed duration , 2009 .

[4]  Hasnaa Zidani,et al.  Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data , 2010, Numerische Mathematik.

[5]  Ellina Grigorieva,et al.  Optimal control of a nonlinear model of economic growth , 2007 .

[6]  G. Leitmann,et al.  Mathematical Methods of Optimal Control , 1971 .

[7]  Shouchuan Hu Differential equations with discontinuous right-hand sides☆ , 1991 .

[8]  Maurizio Falcone,et al.  Numerical Methods for differential Games Based on Partial differential equations , 2006, IGTR.

[9]  N. Subbotina,et al.  On the efficiency of optimal grid synthesis in optimal control problems with fixed terminal time , 2009 .

[10]  Necessary optimality conditions for different phase portraits in a neighborhood of a singular arc , 2006 .

[11]  S. Osher,et al.  High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations , 1990 .

[12]  A. Melikyan Singular Characteristics of First Order PDEs in Optimal Control and Differential Games , 2001 .

[13]  I. Yegorov,et al.  Synthesis of optimal control in a mathematical model of tumour–immune dynamics , 2015 .

[14]  Maurizio Falcone,et al.  Fast Semi-Lagrangian Schemes for the Eikonal Equation and Applications , 2007, SIAM J. Numer. Anal..

[15]  Hasnaa Zidani,et al.  ROC-HJ: Reachability analysis and Optimal Control problems-Hamilton-Jacobi equations , 2013 .

[16]  N. Subbotina,et al.  Optimal synthesis in a control problem with Lipschitz input data , 2008 .

[17]  A. P. Ivanova,et al.  Local solutions of the Hamilton-Jacobi-Bellman equation for some stochastic problems , 2007 .

[18]  H. Schättler Local Fields of Extremals for Optimal Control Problems with State Constraints of Relative Degree 1 , 2006 .

[19]  Ian M. Mitchell,et al.  A Toolbox of Level Set Methods , 2005 .

[20]  U. Ledzewicz,et al.  ANTI-ANGIOGENIC THERAPY IN CANCER TREATMENT AS AN OPTIMAL CONTROL PROBLEM , 2007 .

[21]  M. Kisielewicz Differential Inclusions and Optimal Control , 1991 .

[22]  M. Chyba,et al.  Singular Trajectories and Their Role in Control Theory , 2003, IEEE Transactions on Automatic Control.

[23]  Hasnaa Zidani,et al.  Reachability and Minimal Times for State Constrained Nonlinear Problems without Any Controllability Assumption , 2010, SIAM J. Control. Optim..

[24]  Wang Hai-bing,et al.  High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations , 2006 .

[25]  E. S. Chumerina Choice of optimal strategy of tumor chemotherapy in Gompertz model , 2009 .

[26]  Hasnaa Zidani,et al.  An Efficient Data Structure and Accurate Scheme to Solve Front Propagation Problems , 2010, J. Sci. Comput..

[27]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[28]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[29]  Generalization of Cauchy’s characteristics method to construct smooth solutions to Hamilton-Jacobi-Bellman equations in optimal control problems with singular regimes , 2014 .

[30]  Some algorithms of optimal control , 2006 .

[31]  C. Watanabe,et al.  Optimal Dynamics of Innovation in Models of Economic Growth , 2000 .

[32]  Andrei I. Subbotin,et al.  Generalized solutions of first-order PDEs - the dynamical optimization perspective , 1994, Systems and control.

[33]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[34]  E. Kolpakova,et al.  On the structure of locally Lipschitz minimax solutions of the Hamilton-Jacobi-Bellman equation in terms of classical characteristics , 2010 .

[35]  Urszula Ledzewicz,et al.  AntiAngiogenic Therapy in Cancer Treatment as an Optimal Control Problem , 2007, SIAM J. Control. Optim..

[36]  V. Krotov,et al.  Global methods in optimal control theory , 1993 .

[37]  Boundary singularities and characteristics of Hamilton–Jacobi equation , 2010 .

[38]  Exact solutions of the hamilton-jacobi-bellman equation for problems of optimal correction with a constrained overall control resource , 2004 .

[39]  Alexandre M. Bayen,et al.  Validating a Hamilton-Jacobi Approximation to Hybrid System Reachable Sets , 2001, HSCC.

[40]  Emmanuel Trélat,et al.  Optimal Control with State Constraints and the Space Shuttle Re-entry Problem , 2003 .

[41]  Chi-Wang Shu,et al.  High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes , 2003, SIAM J. Sci. Comput..

[42]  N. Subbotina,et al.  Classical characteristics of the bellman equation in constructions of grid optimal synthesis , 2010 .

[44]  V. F. Borisov,et al.  Singular Optimal Regimes in Problems of Mathematical Economics , 2005 .

[45]  Practical control optimization schemes based on the extension principle , 2006 .

[46]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[47]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[48]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[49]  Pierre Bernhard,et al.  Geometry of Optimal Paths around Focal Singular Surfaces in Differential Games , 2005 .

[50]  A. Bratus,et al.  Solution of the Feedback Control Problem in the Mathematical Model of Leukaemia Therapy , 2013, J. Optim. Theory Appl..

[51]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[52]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[53]  Arik Melikyan,et al.  Generalized characteristics of first order PDEs , 1998 .

[54]  Emiliano Cristiani,et al.  Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach , 2009, 0910.0521.

[55]  S. Osher A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations , 1993 .

[56]  Singular boundary characteristics of the Hamilton–Jacobi equation☆ , 2010 .

[57]  Volker Michel Singular Optimal Control - The State of the Art , 1996 .

[58]  Chihiro Watanabe,et al.  Optimal Trajectories of the Innovation Process and Their Matching with Econometric Data , 2002 .

[59]  A. Manitius Optimization and Nonsmooth Analysis (Frank H. Clarke) , 1985 .

[60]  M. V. Day,et al.  Simple Singularities for Hamilton-Jacobi Equations with Max-Concave Hamiltonians and Generalized Characteristics , 2008 .

[61]  E. Blum,et al.  The Mathematical Theory of Optimal Processes. , 1963 .

[62]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1990, 29th IEEE Conference on Decision and Control.

[63]  N. Subbotina The method of characteristics for Hamilton—Jacobi equations and applications to dynamical optimization , 2006 .

[64]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[65]  A. Bratus,et al.  Smooth solutions of the Hamilton-Jacobi-Bellman equation in a mathematical model of optimal treatment of viral infections , 2010 .