# Introduction to Stochastic Control Theory by Karl J. Astrom

By Karl J. Astrom

This article for upper-level undergraduates and graduate scholars explores stochastic keep watch over idea by way of research, parametric optimization, and optimum stochastic regulate. restricted to linear structures with quadratic standards, it covers discrete time in addition to non-stop time structures. 1970 variation.

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The Center Manifold Theorem is not Remark. 12) for all ~n(x) ~ M. 11) Ft of has dimension E M. (b) n V Y aO' such that: (a) ~(x) denote the belonging to the part of the spectrum on the unit circle; assume that d < Y ~ in the following sense: since k, we get a sequence of center manifolds but their intersection may be empty. 6 ~, THE HOPF BIFURCATION AND ITS APPLICATIONS 20 regarding the differentiability of M. We will be particularly interested in the case in which bifurcation to stable closed orbits occurs.

10) (xu (y) = X*(u(y) ,y». 12) < 1, we get 11D%(y) II < 1 for all y. We shall carry the estimates just one step further. 13) we have IID2~(y)11 <1 for all y. At this point it should be plausible by imposing a sequence of stronger and stronger conditions on I I Djy(y) II < 1 for all y, Y,A, that we can arrange j = 3,4, •.. ,k+1. * * THE HOPF BIFURCATION AND ITS APPLICATIONS 37 The verification that this is in fact possible is left to the reader. e. u = 0, Du yeo) o 0) IL%. (0) o D% (assuming we note that since 0 is a solution of AU(O) + X(u(O) ,0) = 0 ~u(O) 0, By + Y(u(y) ,y) 0 and [A Du(O) + D1X(0,0)Du(0) + D2X(0,0)] ·Dy(O) [A· 0 + 0 + 0]· Dy (0) = O.

Keep the notation and assumptions of the A is SUfficiently small (and if center manifold theorem. If liB-II I is close enough to one), there exists a function defined and k u*, Y, times continuously differentiable on all of with a second-order zero at the origin, such that a) invariant for b) lim n+ oo I Ix n ru* The manifold "xii If As with n u*(y),y E Y} {(x,y) Ix is in the strict sense. 0/ - u * (y = ) II = I IB-li < 0 I, 1, and y is arbitrary then (where (xn'Yn) = we shall treat 0/ n (x,y)).