Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

Viorica Mariela Ungureanu

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 2, page 317-342
  • ISSN: 0011-4642

Abstract

top
In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [chen], for finite dimensional stochastic equations or [UC], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [1990], [ukl]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on 𝐑 + and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [ukl] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [1990]).

How to cite

top

Ungureanu, Viorica Mariela. "Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability." Czechoslovak Mathematical Journal 59.2 (2009): 317-342. <http://eudml.org/doc/37926>.

@article{Ungureanu2009,
abstract = {In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [chen], for finite dimensional stochastic equations or [UC], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [1990], [ukl]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on $\{\mathbf \{R\}\}_\{+\}$ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [ukl] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [1990]).},
author = {Ungureanu, Viorica Mariela},
journal = {Czechoslovak Mathematical Journal},
keywords = {Riccati equation; stochastic uniform observability; stabilizability; quadratic control; tracking problem; Riccati equation; stochastic uniform observability; stabilizability; quadratic control; tracking problem},
language = {eng},
number = {2},
pages = {317-342},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability},
url = {http://eudml.org/doc/37926},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Ungureanu, Viorica Mariela
TI - Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 2
SP - 317
EP - 342
AB - In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [chen], for finite dimensional stochastic equations or [UC], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [1990], [ukl]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ${\mathbf {R}}_{+}$ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [ukl] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [1990]).
LA - eng
KW - Riccati equation; stochastic uniform observability; stabilizability; quadratic control; tracking problem; Riccati equation; stochastic uniform observability; stabilizability; quadratic control; tracking problem
UR - http://eudml.org/doc/37926
ER -

References

top
  1. Barbu, V., Prato, G. Da, Hamilton Jacobi Equations in Hilbert Spaces, Research Notes in Mathematics, 86. Boston-London-Melbourne: Pitman Advanced Publishing Program (1983). (1983) Zbl0508.34001MR0704182
  2. Chen, S., Zhou, Xun YU, 10.1137/S0363012998346578, SIAM J. Control Optimization 39 1065-1081 (2000). (2000) Zbl1023.93072MR1814267DOI10.1137/S0363012998346578
  3. Curtain, R., Pritchard, J., Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences. 8. Berlin-Heidelberg-New York: Springer-Verlag. VII (1978). (1978) Zbl0389.93001MR0516812
  4. Curtain, R., Falb, P., 10.1016/0022-247X(70)90037-5, J. Math. Anal. Appl. 31 434-448 (1970). (1970) Zbl0233.60051MR0261718DOI10.1016/0022-247X(70)90037-5
  5. Dragan, V., Morozan, T., 10.1093/imamci/21.3.323, IMA J. Math. Control Inf. 21 323-344 (2004). (2004) Zbl1060.93019MR2076224DOI10.1093/imamci/21.3.323
  6. Grecksch, W., Tudor, C., Stochastic Evolution Equations, A Hilbert Space Approach. Mathematical Research. 85. Berlin: Akademie Verlag (1995). (1995) Zbl0831.60069MR1353910
  7. Douglas, R., Banach Algebra Techniques in Operator Theory, Pure and Applied Mathematics, 49. New York-London: Academic Press. XVI (1972). (1972) Zbl0247.47001MR0361893
  8. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, Berlin, New York (1983). (1983) Zbl0516.47023MR0710486
  9. Prato, G. Da, Quelques résultats d'existence, unicité et régularité pour une problème de la théorie du contrôle, J. Math. Pures et Appl. 52 (1973), 353-375. (1973) MR0358430
  10. Prato, G. Da, Ichikawa, A., 10.1137/0328019, SIAM. J. Control and Optimization 28 359-381 (1990). (1990) Zbl0692.49006MR1040464DOI10.1137/0328019
  11. Prato, G. Da, Zabczyc, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications. 44. Cambridge, Cambridge University Press. xviii (1992). (1992) MR1207136
  12. Prato, G. Da, Ichikawa, A., 10.1007/BF01443614, Appl. Math. Optimization 18 39-66 (1988). (1988) Zbl0647.93057MR0928209DOI10.1007/BF01443614
  13. Prato, G. Da, Ichikawa, A., 10.1016/0167-6911(87)90023-5, Syst. Control Lett. 9 165-172 (1987). (1987) Zbl0678.93051MR0906236DOI10.1016/0167-6911(87)90023-5
  14. Pritchard, A. J., Zabczyc, J., 10.1137/1023003, SIAM Rev. 23 25-52 (1981). (1981) MR0605439DOI10.1137/1023003
  15. Morozan, T., Stochastic uniform observability and Riccati equations of stochastic control, Rev. Roum. Math. Pures Appl. 38 771-781 (1993). (1993) Zbl0810.93069MR1262989
  16. Morozan, T., On the Riccati Equation of Stochastic Control, Optimization, optimal control and partial differential equations. Proc. 1st Fr.-Rom. Conf., Iasi/Rom. (1992). (1992) 
  17. Morozan, T., Linear quadratic, control and tracking problems for time-varying stochastic differential systems perturbed by a Markov chain, Rev. Roum. Math. Pures Appl. 46 783-804 (2001). (2001) Zbl1078.93574MR1929525
  18. Ungureanu, V. M., Riccati equation of stochastic control and stochastic uniform observability in infinite dimensions, Barbu, Viorel (ed.) et al., Analysis and optimization of differential systems. IFIP TC7/WG 7.2 international working conference, Constanta, Romania, September 10-14, 2002. Boston, MA: Kluwer Academic Publishers 421-432 (2003). (2003) Zbl1071.93014MR1993734
  19. Ungureanu, V. M., Uniform exponential stability for linear discrete time systems with stochastic perturbations in Hilbert spaces, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. 7 757-772 (2004). (2004) MR2101664
  20. Ungureanu, V. M., Representations of mild solutions of time-varying linear stochastic equations and the exponential stability of periodic systems, Electron. J. Qual. Theory Differ. Equ. 2004, Paper No. 4, 22 p. (2004). (2004) Zbl1072.60047MR2039027
  21. Ungureanu, V. M., Cost of tracking for differential stochastic equations in Hilbert spaces, Stud. Univ. Babeş-Bolyai, Math. 50 73-81 (2005). (2005) Zbl1113.60061MR2247532
  22. Ungureanu, V. M., 10.1016/j.jmaa.2008.01.058, J. Math. Anal. Appl. 343 446-463 (2008). (2008) Zbl1147.60043MR2412142DOI10.1016/j.jmaa.2008.01.058
  23. Ungureanu, V. M., 10.1080/14689360802275773, Dynamical Systems 23 333-350 (2008). (2008) Zbl1155.93017MR2455264DOI10.1080/14689360802275773
  24. Tudor, C., 10.1137/0328014, SIAM J. Control Optimization 28 253-264 (1990). (1990) Zbl0693.93086MR1040459DOI10.1137/0328014
  25. Yosida, K., Functional analysis, 6th ed. Grundlehren der mathematischen Wissenschaften, 123. Berlin-Heidelberg-New York: Springer-Verlag. XII (1980). (1980) Zbl0435.46002MR0617913

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.