By Oswaldo Luiz do Valle Costa

1.Introduction.- 2.A Few instruments and Notations.- 3.Mean sq. Stability.- 4.Quadratic optimum keep watch over with whole Observations.- 5.H2 optimum regulate With whole Observations.- 6.Quadratic and H2 optimum regulate with Partial Observations.- 7.Best Linear filter out with Unknown (x(t), theta(t)).- 8.H_$infty$ Control.- 9.Design Techniques.- 10.Some Numerical Examples.- A.Coupled Differential and Algebraic Riccati Equations.- B.The Adjoint Operator and a few Auxiliary Results.- References.- Notation and Conventions.- Index

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**Example text**

1. Re{λ(L)} < 0. Re{λ(A)} < 0. 2 with q = 0 and Q = 0. 38) . 7 we have that Q(t) = L(Q(t)). 39) 0 for arbitrary initial condition ϑ0 . Suppose now that (i) holds, so that, for arbitrary ∞ initial condition ϑ0 , we have that 0 E( x(t) 2 ) dt < ∞, and consider any H = n+ (H1 , . . , HN ) ∈ HC . 4 Mean-Square Stability for the Homogeneous Case 45 [234], Chap. √We take independent variables x0i and θ0 with the following distribution: x0i = N nλk (Hi )ek (Hi ) with probability n1 for k = 1, . . , n and θ (0) = i with probability N1 for i = 1, .

470). 7) with initial condition x(t0 ) = x0 , and suppose that Assumptions (A1) and (A2) are satisfied. Then: (i) for each (t0 , x0 ) ∈ R+ × Rn , there exists a continuous function φ : R+ → Rn ˙ = p(φ(t), t); such that φ(t0 ) = x0 , and for all t ∈ R+ \ D, we have that φ(t) (ii) this function is unique. 9) where A, B, and u are of class PC. We have that for the global Lipschitz condition, A(t)x + B(t)u(t) − A(t)y + B(t)u(t) = A(t)(x − y) ≤ A(t) x−y , and thus Assumption (A2) is satisfied with k(t) = A(t) .

23 that Re{λ(A )} = Re{λ(A)} < 0. 15 that (a) is equivalent to mean-square stability. The fact that R and C may be interchanged is proven as follows. Obviously, the existence of Gj ∈ B(Rn ) n in (b) is sufficient for Gj ∈ B(Cn ). The necessity is due to A√ i ∈ B(R ) for all R I i ∈ S. In this case, whenever (b) is true, we have G = G + −1G for some R I I I n GR = (GR 1 , . . , GN ) and G = (G1 , . . , GN ) in H . 4 Mean-Square Stability for the Homogeneous Case 51 transpose we have (bearing in mind that the entries of Ai are real) √ √ Li GR + −1Li GI < 0, Li GR − −1Li GI < 0, i ∈ S, so that, summing both expressions, we obtain the real version of (b).