Quadratic integral games and causal synthesis

Author:
Yuncheng You

Journal:
Trans. Amer. Math. Soc. **352** (2000), 2737-2764

MSC (1991):
Primary 90D25, 49N35; Secondary 45D05, 47N70, 49N55, 93B36

DOI:
https://doi.org/10.1090/S0002-9947-99-02457-5

Published electronically:
October 21, 1999

MathSciNet review:
1650054

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The game problem for an input-output system governed by a Volterra integral equation with respect to a quadratic performance functional is an untouched open problem. In this paper, it is studied by a new approach called projection causality. The main result is the causal synthesis which provides a causal feedback implementation of the optimal strategies in the saddle point sense. The linear feedback operator is determined by the solution of a Fredholm integral operator equation, which is independent of data functions and control functions. Two application examples are included. The first one is quadratic differential games of a linear system with arbitrary finite delays in the state variable and control variables. The second is the standard linear-quadratic differential games, for which it is proved that the causal synthesis can be reduced to a known result where the feedback operator is determined by the solution of a differential Riccati operator equation.

**1.**M. Green and D. J. N. Limebeer,*Linear Robust Control*, Prentice-Hall, 1995.**2.**A. Ichikawa,*Linear quadratic differential games in Hilbert spaces*, SIAM J. Control and Optimization**14**(1976), 120-136. MR**52:16681****3.**J.-L. Lions,*Some Methods in the Mathematical Analysis of Systems and Their Control*, Science Press, Beijing, 1982. MR**84m:49003****4.**E. B. Lee and Y. You,*Optimal syntheses for infinite dimensional linear delayed state-output systems: a semicausality approach*, Appl. Math. and Optimization**19**(1989), 113-136. MR**89i:93050****5.**E. B. Lee and Y. You,*Quadratic optimization for infinite dimensional linear differential difference type systems: synthesis via the Fredholm equations*, SIAM J. Control and Optimization**28**(1990), 265-293. MR**91a:49014****6.**D. L. Lukes and D. L. Russell,*A global theory for linear quadratic differential games*, J. Math. Anal. Appl.**33**(1979), 96-123. MR**42:4220****7.**A. Pazy,*Semigroups of Linear Operators and Applications to Partial Differential equations*, Springer-Verlag, New York, 1983. MR**85g:47061****8.**A. J. Pritchard and Y. You,*Causal feedback optimal control for Volterra integral equations*, SIAM J. Control and Optimization**34**(1996), 1874-1890. MR**97k:49035****9.**Y. You,*Closed-loop syntheses for quadratic differential games of distributed systems*, Chinese Ann. Math.**6B**(1985), 325-334. MR**87k:90318****10.**Y. You,*Quadratic differential game of general linear delay systems*, J. Math. Pures Appl.**69**(1990), 261-283. MR**92e:90144**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
90D25,
49N35,
45D05,
47N70,
49N55,
93B36

Retrieve articles in all journals with MSC (1991): 90D25, 49N35, 45D05, 47N70, 49N55, 93B36

Additional Information

**Yuncheng You**

Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620

Email:
you@math.usf.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02457-5

Keywords:
Volterra integral equation,
quadratic game,
optimal strategy,
projection causality,
output feedback,
Fredholm operator equation.

Received by editor(s):
April 29, 1996

Received by editor(s) in revised form:
April 1, 1998

Published electronically:
October 21, 1999

Article copyright:
© Copyright 2000
American Mathematical Society