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where using software touse barcode standards 128 on web,windows application USS Code 93 x ( ). is the optimum state x (t) = H p = f (t, x (t), u Code 128 Code Set A for None (t)) , x (0) = x0 ,. and the co-state p solves the retrograde differential equation q(x (T )) H(t, p(t), x (t), u (t)) , pT (T ) = x x The minimum principle applies to a broader class of optimal control problems where the solution need not be smooth (for example, the optimal open-loop control could be of the bang-bang type, switching back and forth between extreme values), but when the solution is smooth, and the HJB equation admits a continuously differentiable solution, the co-state variable is equal to the gradient of the optimal value function evaluated on the optimal trajectory, that is pT (t) = pT (t) = V (t, x (t))/ x ,. If there are any Code 128A for None pointwise constraints on u for each t, such as u(t) U(t), where U(t) is a given constraint set in Rr , then the minimization below will be over this set U(t).. Optimization, Game Theory, and Optimal & Robust Control which is the connection between the HJB approach to optimal control and the minimum principle. A.3.4.

H -Optimal Control We discus in this subsection a robust control approach to optimal control, rst when the state dynamics are disturbed by an unknown input (so called disturbance input), and then when also the measurements are perturbed by disturbance (or unknown noise). Hence, the earlier state dynamics are now replaced by x = f (t; x(t), u(t), w(t)) , x(0) = x0 , t 0 ,. where w( ) is the Software Code128 unknown disturbance input, of dimension not necessarily the same as that of u or that of x. The cost functional is still given as before. L(u, w) =. g(t, x(t), u(t)) Code 128 Code Set A for None dt + q(x(T )) ,. where now we have w also appearing as an argument because L depends on w through x. The objective is to nd the controller (say, state-feedback controller) under which some safe value of the performance is obtained regardless of the choice of w. One criterion is to make the ratio of L to the square of the norm of w as small as possible for worst (maximizing) values of w.

For mathematical convenience, one also lumps the norm of x0 with that of w, by viewing x0 as also unknown, leading to the ratio L(u, w)/[ w where w. 2 T 2. + qo (x0 )]. wT (t)w(t) dt ,. qo (x0 ) is a pos itive cost on x0 , which is zero at zero, such as the square of the Euclidean norm. We want to nd the state-feedback control law which minimizes the maximum of the ratio over all (unrestricted) w s and xo s. If that min max, or more precisely the sup inf value is denoted by , even though it may not be achieved, one would be looking at feedback controls that achieve (for the ratio) some > , known as disturbance attenuation.

In the linear-quadratic case (that is when the dynamics are linear and L is quadratic, supremum (over all w s) of this ratio (or of its square root) is known as the H norm, and hence the controller sought would be minimizing the H norm the reason why this class of problems is known as H optimal control or simply H control. An application of H optimal control to network security is presented in 7. It can be shown that minimization of this H norm is equivalent to solving a game (a zero-sum differential game), with dynamics as given above, with a parametrized objective function L (u, w) = L(u, w) 2 w.

2 q0 (x0 ) ,. where the scalar > , and we take x0 at this point to be xed and known (instead of being controlled by an adversary). This objective function will be maximized by the dis-.
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