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Lemma 1. Let (Ae,Be,Ce,De) be given and assume there exists a n n x w matrix 2, satisfying Jef^T (46) and 0 = Ä1 + £ΑΎ + y~2lRl + V. (47) OPTIMAL REDUCED-ORDER SUBSPACE-OBSERVER DESIGN 33 Then, (A9 D) is stabilizable (48) Ae is asymptotically stable. (49) if and only if Furthermore, in this case, (50) \\H(s)\L

La,b, and c. In Table I the number εΡ is defined by ερ ;= [ΡΔ'*(0, u*)- PAh(0, Μ Ρ Ν )]/Ρ Δ '"(0, UPN). (76) The figures represent the typical form of PAh(t; u), t e [0, T ] , u = u*, wPN. VI. CONCLUSIONS Define μ:= max (Ρ Δ ' Λ (ί,M*) - PAJl(t,ΜΡΝ))/ΡΔ'Α(ί; wPN). (77) 0<ί<Γ The numerical results suggest that μ = εΡ. 05. The implementation of the pursuit strategy (ν$ί9 νjf2) consists of two stages. In the first stage, which is a prelaunch stage, Eqs. }, and N A = T for some N > 0. 05, the guidance law (v™9 v™) to be a good approximation to an optimal feedback pursuit strategy.

Control AC-33, 591-595 (1988). 19. Y. HALEVI, "The Optimal Reduced-Order Estimator for Systems with Singular Measurement Noise," IEEE Trans. Autom. Control 34 (1989). 20. W. M. WONHAM, "Linear Multivariable Control: A Geometric Approach," SpringerVerlag, New York, 1979. CONTROL AND DYNAMIC SYSTEMS, VOL. 32 PROPORTIONAL NAVIGATION AND THE GAME OF TWO CARS IN THREE DIMENSIONS: A TIME-DEPENDENT STOCHASTIC DIFFERENTIAL GAME YAAKOV YAVIN ROSALIE DE VILLIERS Centre for Advanced Computing and Decision Support Council for Scientific and Industrial Research Pretoria 0001, South Africa I.

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