Fabbri, Giorgio (2007): Viscosity solutions to delay differential equations in demoeconomy.
This is the latest version of this item.

PDF
MPRA_paper_5117.pdf Download (359Kb)  Preview 
Abstract
Economic and demographic models governed by linear delay differential equations are expressed as optimal control problems in infinite dimensions. A general objective function is considered and the concavity of the Hamiltonian is not required. The value function is a viscosity solution of the HamiltonJacobiBellman (HJB) equation and a verification theorem is proved.
Item Type:  MPRA Paper 

Original Title:  Viscosity solutions to delay differential equations in demoeconomy 
Language:  English 
Keywords:  viscosity solutions; delay differential equation; vintage models 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C61  Optimization Techniques; Programming Models; Dynamic Analysis M  Business Administration and Business Economics; Marketing; Accounting > M3  Marketing and Advertising > M30  General O  Economic Development, Technological Change, and Growth > O4  Economic Growth and Aggregate Productivity > O41  One, Two, and Multisector Growth Models 
Item ID:  5117 
Depositing User:  Giorgio Fabbri 
Date Deposited:  02. Oct 2007 
Last Modified:  15. Feb 2013 03:01 
References:  Bachar, M. and Dorfmayr, A. (2004). Hiv treatment models with time delay. Comptes Rendus Biologies, 327: 983 994. Baker, C., Bocharov, G. and Rihan, F. (1999). A report on the use of delay differential equations in numerical modelling in the biosciences. MCCM Technical Report, Vol. 343. Bensoussan, A., Da Prato, G., Delfour, M. C. and Mitter, S. K. (1992). Representation and control of in nitedimensional systems. Vol. 1. Systems & Control: Foundations & Applications. Boston, MA: Birkhäuser Boston Inc. Bocharova, G. and Rihanb, F. (2000). Numerical modelling in biosciences using delay differential equations. Journal of Computational and Applied Mathematics, 125: 183 199. Boucekkine, R., de la Croix, D. and Licandro, O. (2002). Vintage human capital, demographic trends, and endogenous growth. Journal of Economic Theory, 102(2): 340 375. Boucekkine, R., de la Croix, D. and Licandro, O. (2004). Modelling vintage structures with ddes: principles and applications.. Mathematics for Population Studies, 11(34): 151 179. Boucekkine, R., del Rio, F. and Martinez, R. (2006). A vintage ak theory of obsolescence and depreciation. work in progress. Boucekkine, R., Germain, M., Licandro, O. and Magnus, A. (2001). Numerical solution by iterative methods of a class of vintage capital models. Journal of Economic Dynamics, 25: 655 699. Boucekkine, R., Licandro, O. and Paul, C. (1997). Differentialdifference equations in economics: on the numerical solution of vintage capital growth models. Journal of Economic Dynamics and Control, 21: 347 362. Boucekkine, R., Licandro, O., Puch, L. A. and del Rio, F. (2005). Vintage capital and the dynamics of the AK model. Journal of Economic Theory, 120(1): 39 72. Cannarsa, P., Gozzi, F. and Soner, H. M. (1991). A boundary value problem fo HamiltonJacobi equations in Hilbert spaces. Applied Mathematics and Optimization, 24(2): 197 220. Cannarsa, P., Gozzi, F. and Soner, H. M. (1993). A dynamic programming approach to nonlinear boundary control problems of parabolic type. Journal of Functional Analysis, 117(1): 25 61. Cannarsa, P. and Tessitore, M. E. (1994). Cauchy problem for the dynamic programming equation of boundary control. in `Boundary control and variation (Sophia Antipolis, 1992)'. Vol. 163 of Lecture Notes in Pure and Applied Mathematics. Dekker. New York. pp. 13 26. Cannarsa, P. and Tessitore, M. E. (1996a). Cauchy problem for HamiltonJacobi equations and Dirichlet boundary control problems of parabolic type. in `Control of partial differential equations and applications (Laredo, 1994)'. Vol. 174 of Lecture Notes in Pure and Applied Mathematics. Dekker. New York. pp. 31 42. Cannarsa, P. and Tessitore, M. E. (1996b). In nitedimensional HamiltonJacobi equations and Dirichlet boundary control problems of parabolic type. SIAM Journal on Control and Optimization, 34(6): 1831 1847. Crandall, M. G. and Lions, P.L. (1983). Viscosity solutions of HamiltonJacobi equations. Transactions of the American Mathematical Society, 277(1): 1 42. Crandall, M. G. and Lions, P.L. (1985). HamiltonJacobi equations in infinite dimensions. I. Uniqueness of viscosity solutions. Journal of Functional Analysis, 62(3): 379 396. Crandall, M. G. and Lions, P.L. (1986a). HamiltonJacobi equations in infinite dimensions. II. Existence of viscosity solutions. Journal of Functional Analysis, 65(3): 368 405. Crandall, M. G. and Lions, P.L. (1986b). HamiltonJacobi equations in infinite dimensions. III. Journal of Functional Analysis, 68(2): 214 247. Crandall, M. G. and Lions, P.L. (1990). Viscosity solutions of HamiltonJacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms. Journal of Functional Analysis, 90(2): 237 283. Crandall, M. G. and Lions, P.L. (1991). Viscosity solutions of HamiltonJacobi equations in infinite dimensions. V. Unbounded linear terms and Bcontinuous solutions. Journal of Functional Analysis, 97(2): 417 465. Crandall, M. G. and Lions, P.L. (1994a). HamiltonJacobi equations in infinite dimensions. VI. Nonlinear A and Tataru's method refined. in `Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991)'. Vol. 155 of Lecture Notes in Pure and Applied Mathematics. Dekker. New York. pp. 51 89. Crandall, M. G. and Lions, P.L. (1994b). Viscosity solutions of HamiltonJacobi equations in infinite dimensions. VII. The HJB equation is not always satisfied. Journal of Functional Analysis, 125(1): 111 148. Culshaw, R. and Ruan, S. (2000). A delaydifferential equation model of hiv infection of cd4+ tcells. Mathematical Biosciences, 165: 27 39. Deissenberg, C., Feichtinger, G., Semmler, W. and Wirl, F. (2004). History dependent multiple equilibria and global dynamics in efficient intertemporal optimization models. in W. Barnett, C. Deissenberg and G. E. Feichtinger, eds, `Economic Complexity: Nonlinear Dynamics, Multiagents Economies, and Learning'. Vol. 14 of ISETE. Elsevier, Amsterdam. pp. 91 122. Delfour, M. C. (1980). Status of the state space theory of linear hereditary differential systems with delays in state and control variables. in `Analysis and optimization of systems (Proc. Fourth Internat. Conf., Versailles, 1980)'. Vol. 28 of Lecture Notes in Control and Information Sciences. Springer. Berlin. pp. 83 96. Delfour, M. C. (1984). Linear optimal control of systems with state and control variable delays. Automatica  IFAC, 20(1): 69 77. Delfour, M. C. (1986). The linear quadratic optimal control problem with delays in the state and control variables: a state space approch. SIAM journal of control and optimization, 24: 835 883. Fabbri, G. (2006a). A Dynamic Programming approach in Hilbert spaces for a family of applied delay optimal control problems. in R. Glowinski and J. Zolesio., eds, `Free and Moving Boundaries: Analysis, Simulation and Control'. to appear. Fabbri, G. (2006b). A viscosity solution approach to the infinite dimensional hjb related to boundary control problem in transport equation. submitted. Fabbri, G., Faggian, S. and Gozzi, F. (2006). On the Dynamic Programming approach to economic models governed by DDE's. submitted. Fabbri, G. and Gozzi, F. (2006). Vintage Capital in the AK growth model: a Dynamic Programming approach. submitted. Faggian, S. (2001/2002). First order HamiltonJacobi equations in hilbert spaces, boundary optimal control and applications to economics. PhD thesis. Universitá degli studi di Pisa. Faggian, S. (2005a). Boundary control problems with convex cost and dynamic pro gramming in in nite dimension. II. Existence for HJB. Discrete and Continuous Dynamical Systems, 12(2): 323 346. Faggian, S. (2005b). Regular solutions of firstorder HamiltonJacobi equations for boundary control problems and applications to economics. Applied Mathematics and Optimization, 51(2): 123 162. Faggian, S. and Gozzi, F. (2004). On the dynamic programming approach for optimal control problems of PDE's with age structure. Mathematical Population Studies, 11(34): 233 270. Feichtinger, G., Hartl, R. and Sethi, S. (1994). Dynamical optimal control models in advertising: Recent developments. Management Science, 40: 195 226. Fleming, W. H. and Rishel, R. W. (1975). Deterministic and stochastic optimal control.. Applications of Mathematics. Vol. 1. BerlinHeidelbergNew York: SpringerVerlag. XIII, 222 p. Gozzi, F. and Marinelli, C. (2004). `Optimal advertising under uncertainty with memory and lags'. mimeo. to appear in the proceedings of the conference "SPDE and Applications" 2004. Gozzi, F., Marinelli, C. and Savin, S. (2006). Optimal advertising under uncertainty with carryover e ects. Gozzi, F., Sritharan, S. S. and Swiech, A. (2002). Viscosity solutions of dynamicprogramming equations for the optimal control of the twodimensional Navier Stokes equations. Archive for Rational Mechanics and Analysis, 163(4): 295 327. Hethcote, H. and van den Driessche, P. (1995). An sis epidemic model with variable population size and a delay. Journal of Mathematical Biology, 34: 177 194. Hethcote, H. and van den Driessche, P. (2000). Two sis epidemiologic models with delays. Journal of Mathematical Biology, 40: 3 26. Ishii, H. (1993). Viscosity solutions of nonlinear secondorder partial differential equations in Hilbert spaces. Communications in Partial Differential Equations, 18(34): 601 650. Li, X. J. and Yong, J. M. (1995). Optimal control theory for infinitedimensional systems. Systems & Control: Foundations & Applications. Boston, MA. Birkhäuser Boston Inc. Luzyanina, T., Roose, D. and Engelborghs, K. (2004). Numerical stability analysis of steady state solutions of integral equations with distributed delays. Applied numerical mathematics, 50: 72 92. Nerlone, M. and Arrow, J. (1962). Optimal advertising policy under dynamic conditions. Economica, New Series, 29(114): 129 142. Pazy, A. (1983). Semigroups of linear operators and applications to partial differential equations.. Applied Mathematical Sciences, 44. SpringerVerlag, New York. Smith, H. (1983). Hopf bifurcation in a system of functional equations modeling the spread of an infectious disease. SIAM Journal on Applied Mathematics, 43(2): 370 385. Tataru, D. (1992a). Viscosity solutions for the dynamic programming equations. Applied Mathematics and Optimization, 25(2): 109 126. Tataru, D. (1992b). Viscosity solutions of HamiltonJacobi equations with unbounded nonlinear terms. Journal of Mathematical Analysis and Applications, 163(2): 345 392. Tataru, D. (1994). Viscosity solutions for HamiltonJacobi equations with unbounded nonlinear term: a simpli ed approach. Journal of Di erential Equations, 111(1): 123 146. Vinter, R. B. and Kwong, R. H. (1981). The infinite time quadratic control problem for linear system with state control delays: An evolution equation approach. SIAM journal of control and optimization, 19: 139 153. Waltman, P. (1974). Deterministic threshold models in the theory of epidemics. In `Lecture Notes Biomath.'. Vol. 1. Berlin: Springer. Yong, J. and Zhou, X. Y. (1999). Stochastic controls. Hamiltonian systems and HJB equations.. Applications of Mathematics. 43. New York, NY: Springer. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/5117 
Available Versions of this Item

Viscosity solutions approach to economic models governed by DDEs. (deposited 19. Apr 2007)
 Viscosity solutions to delay differential equations in demoeconomy. (deposited 02. Oct 2007) [Currently Displayed]