Linear Quadratic ZeroSum TwoPerson Differential Games
Abstract
As in optimal control theory, linear quadratic (LQ) differential games (DG) can be solved, even in high dimension, via a Riccati equation. However, contrary to the control case, existence of the solution of the Riccati equation is not necessary for the existence of a closedloop saddle point. One may “survive” a particular, nongeneric, type of conjugate point. An important application of LQDGs is the socalled H _{ ∞ }optimal control, appearing in the theory of robust control.
Perfect State Measurement
Theorem 1

A sufficient condition for the existence of a closedloop saddle point, then given by \({(\varphi }^{\star }{,\psi }^{\star })\) in (3) , is that Eq. (1) has a solution P(t) defined over [t _{0},T].

A necessary and sufficient condition for the existence of an openloop saddle point is that Eq. (2) has a solution over [t _{0} ,T] (and then so does (1)). In that case, the pairs \((\hat{u}(\cdot ),\hat{v}(\cdot ))\),\((\hat{u}(\cdot ){,\psi }^{\star })\) , and \({(\varphi }^{\star }{,\psi }^{\star })\) are saddle points.

A necessary and sufficient condition for \({(\varphi }^{\star },\hat{v}(\cdot ))\) to be a saddle point is that Eq. (1) has a solution over [t _{0},T].

In all cases where a saddle point exists, the Value function is \(V (t,x) =\ x\_{P(t)}^{2}\).
Theorem 2
 1.
\(x_{0} \in \mathcal{R}(X(t_{0}))\).
 2.
For almost all \(t \in [t_{0},T]\),\(\mathcal{R}(D(t)) \subset \mathcal{R}(X(t))\).
 3.
\(\forall t \in [t_{0},T]\),\(Y (t){X}^{\dag }(t) \geq 0\).
In a case where X(t) is only singular at an isolated instant t ^{⋆} (then conditions 1 and 2 above are automatically satisfied), called a conjugate point but where YX ^{− 1} remains positive definite on both sides of it, the conjugate point is called even. The feedback gain F = − R ^{− 1} B ^{ t } P diverges upon reaching t ^{⋆}, but on a trajectory generated by this feedback, the control u(t) = F(t)x(t) remains finite. (See an example in Bernhard 1979.)
If T = ∞, with all system and payoff matrices constant and Q > 0, Mageirou (1976) has shown that if the algebraic Riccati equation obtained by setting \(\dot{P} = 0\) in (1) admits a positive definite solution P, the game has a Value \(\x\_{P}^{2}\), but (3) may not be a saddle point. (ψ ^{⋆} may not be an equilibrium strategy.)
H _{ ∞ }Optimal Control
This subsection is entirely based upon Başar and Bernhard (1995). It deals with imperfect state measurement, using Bernhard’s nonlinear minimax certainty equivalence principle (Bernhard and Rapaport 1996).
Several problems of robust control may be brought to the following one: a linear, timeinvariant system with two inputs (control input \(u \in {\mathbb{R}}^{m}\) and disturbance input \(w \in {\mathbb{R}}^{\ell}\)) and two outputs (measured output \(y \in {\mathbb{R}}^{p}\) and controlled output \(z \in {\mathbb{R}}^{q}\)) is given. One wishes to control the system with a nonanticipative controller u(⋅) = ϕ(y(⋅)) in order to minimize the induced linear operator norm between spaces of squareintegrable functions, of the resulting operator w(⋅)↦z(⋅).
We shall extend somewhat this classical problem by allowing either a time variable system, with a finite horizon T, or a timeinvariant system with an infinite horizon.
Finite Horizon
Theorem 3
 1.The following Riccati equation has a solution over [t _{0} ,T] :$$\dot{P} = PA + {A}^{t}P  (PB + S){R}^{1}({B}^{t}P + {S}^{t}) {+\gamma }^{2}PMP + Q\,,\quad P(T) = K\,.$$(8)
 2.The following Riccati equation has a solution over [t _{0} ,T] :$$\dot{\Sigma } = A\Sigma + \Sigma {A}^{t}  (\Sigma {C}^{t} + L){N}^{1}(C\Sigma + {L}^{t}) {+\gamma }^{2}\Sigma Q\Sigma + M\,,\quad \Sigma (t_{ 0}) = \Sigma _{0}\,.$$(9)
 3.The following spectral radius condition is satisfied:$$\forall t \in [t_{0},T]\,,\quad \rho (\Sigma (t)P(t)) {<\gamma }^{2}\,.$$(10)
Infinite Horizon
The infinite horizon case is the traditional H _{ ∞ }optimal control problem reformulated in a state space setting. We let all matrices defining the system be constant. We take the integral in (7) from − ∞ to + ∞, with no initial or terminal term of course. We add the hypothesis that the pairs (A, B) and (A, D) are stabilizable and the pairs (C, A) and (H, A) detectable. Then, the theorem is as follows:
Theorem 4
\(\gamma {\leq \gamma }^{\star }\) if and only if the following conditions are satisfied: The algebraic Riccati equations obtained by placing \(\dot{P} = 0\) and \(\dot{\Sigma } = 0\) in (8) and (9) have positive definite solutions, which satisfy the spectral radius condition (10) . The optimal controller is given by Eqs. (11) and (12) , where P and Σ are the minimal positive definite solutions of the algebraic Riccati equations, which can be obtained as the limit of the solutions of the differential equations as t →−∞ for P and t →∞ for Σ.
Conclusion
The similarity of the H _{ ∞ }optimal control theory with the LQG, stochastic, theory is in many respects striking, as is the duality observation control. Yet, the “observer” of H _{ ∞ }optimal control does not arise from some estimation theory but from the analysis of a “worst case.” The best explanation might be in the duality of the ordinary, or (+, ×), algebra with the idempotent (max, + ) algebra (see Bernhard 1996). The complete theory of H _{ ∞ }optimal control in that perspective has yet to be written.
CrossReferences
Generalized FiniteHorizon Linear Quadratic Control Optimal Control
Linear Quadratic Optimal Control
PursuitEvasion Games and ZeroSum TwoPerson Differential Games
Stochastic Linearquadratic control
Game Theory: Historical Overview
Optimization based Robust Control
Robust control of infinite dimensional systems
Robust H2 Performance in Feedback Control
Bibliography
 Başar T, Bernhard P (1995) H ^{∞}optimal control and related minimax design problems: a differential games approach, 2nd edn. Birkhäuser, BostonGoogle Scholar
 Bernhard P (1979) Linear quadratic zerosum twoperson differential games: necessary and sufficient condition. JOTA 27(1):51–69CrossRefzbMATHMathSciNetGoogle Scholar
 Bernhard P (1980) Linear quadratic zerosum twoperson differential games: necessary and sufficient condition: comment. JOTA 31(2):283–284CrossRefzbMATHMathSciNetGoogle Scholar
 Bernhard P (1996) A separation theorem for expected value and feared value discrete time control. COCV 1:191–206CrossRefzbMATHGoogle Scholar
 Bernhard P, Rapaport A (1996) Minmax certainty equivalence principle and differential games. Int J Robust Nonlinear Control 6:825–842CrossRefzbMATHMathSciNetGoogle Scholar
 Delfour M (2005) Linear quadratic differential games: saddle point and Riccati equation. SIAM J Control Optim 46:750–774CrossRefMathSciNetGoogle Scholar
 Ho YC, Bryson AE, Baron S (1965) Differential games and optimal pursuitevasion strategies. IEEE Trans Autom Control AC10:385–389CrossRefMathSciNetGoogle Scholar
 Mageirou EF (1976) Values and strategies for infinite time linear quadratic games. IEEE Trans Autom Control AC21:547–550CrossRefMathSciNetGoogle Scholar
 Zhang P (2005) Some results on zerosum twoperson linear quadratic differential games. SIAM J Control Optim 43:2147–2165CrossRefGoogle Scholar