By Larry Davis, Don Greer

Planes Names & Dames Vol three

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PROBLEM (Q 1). Minimize the functional I=max F, I O~t~l (23a) subject to the dynamical constraints (20) and the boundary conditions (21)-(22). In (23a), p denotes the density, p = p(h). Invoking (7)-(8), (23a) can be rewritten as (23b) O~t~l (23c) 27 Minimax Optimal Control PROBLEM (Q2). Minimize the functional O:s; t:s; 1 [=max F, t (24a) subject to the dynamical constraints (20) and the boundary conditions (21}-(22). Invoking (7}-(8), (24a) can be rewritten as [=F* (24b) F* _vPV3~0, (24c) We note that, except for a proportionality constant, the minimax function F in (23) represents the dynamic pressure and the minimax function F in (24) represents an approximation to the stagnation-point heating rate.

O. A. 21 22 A. Miele and P. Venkataraman index. This problem can be formulated in the following form, called Problem (P) for easy identification. PROBLEM (P). Minimize the functional 1= f f(x, u, 7T', t)dt + [h(x, 7T')]o+ [g(x, 7T')]1 (1) with respect to the state x(t), the control u(t), and the parameter 7T', which satisfy the following constraints: i = S(x, 4> (x, U, 7T', t), U, 0,;;; 7T', t) = 0, [w(x, 7T')]o = [t/I(x, 7T')h = t,;;; 1 (2) (3) ° ° (4) (5) In the above equations, the functions f, h, g are scalar, and the functions 4>, S, w, t/I are vectors of appropriate dimensions.

In fact, usually this contribution is important. We are thus motivated to introduce as follows: a Fr j = 1,2, ... 14) Singular Perturbation for Stiff Equations 13 Physically, Fr is the discrepancy between the exact FY and its value under the quasi-equilibrium approximations. Hence, Fr is now known to be small and unimportant. 16b) i=1 = L S';(Y)F. 16c) r=1 and s'; = [ j Eliminating F? br;o] . S, (y), r = 1,2, ... 17) where 00 n dO? Ok = ~ 'T1kdt, i, k = 1,2, ... 18) Since the eigenvalues of 'Tlk are all expected to be small, Or:: and Fr:: and (;00 are now theoretically small and are therefore unimportant.