A practical approach to linear algebra by Choudhary P.

By Choudhary P.

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Let ln Cρ0 + ln Cδ0−n 0 P + I − P + 2, n 0 . 25) that for all N ≥ N0 , h ∈ H , y(N T ; 0, h, L (Ph)) ≤ Cρ0 δ0N (Cδ0−n 0 P + I − P ) h ≤ δ0 h . Step 3. 26) where B(s) Z is the restriction of B(s) on the subspace Z . For each ε > 0, define the cost functional: N T,ε,Z (u) = Jt,h NT ε u(s) t 2 U ds + y NZ (N T ; t, h, u) 2 , u ∈ L 2 (t, N T ; Z ). 27) Then consider the following LQ problem N T,ε,Z (L Q)t,h : W N T,ε,Z (t, h) inf u∈L 2 (t,N T ;Z ) N T,ε,Z Jt,h (u). 17). 25), W N T,ε,Z (0, h) ≤ δ0 h for all h ∈ H, when N ≥ N0 and ε ∈ (0, ε0 ].

110) is exponentially stable. This completes the proof. Miscellaneous Notes The LQ theory is an important field in the control theory. 4 of [8]). In 1960, the author of [43] introduced the relation between LQ problems and optimal feedback controls. The above-mentioned two monographs deal with finite dimensional systems. In the mid-1960s, the LQ theory was extended to partial differential equations in [61, 62]. The LQ theory for general evolution equations with bounded control operators was introduced in [66] and [26] in 1969 and 1976 respectively.

5) for some k ∈ N. Then there is u ∈ L 2 (0, ∞; Z ) so that kT P Pkh = P Φ(kT, s)B(s)u(s)ds, 0 from which, it follows that Py(kT ; 0, h, −u) = 0. Let uˆ = −χ[0,kT ] u. Then uˆ and y(kT ; 0, h, u) ˆ are in L 2 (t, ∞; Z ) and H2 respectively. 4, yield that y(s; 0, h, u) ˆ = y(s; kT, y(kT ; 0, h, u), ˆ u) ˆ = y(s; kT, y(kT ; 0, h, u), ˆ 0) = Φ(s, kT )y(kT ; 0, h, u) ˆ −ρ(s−kT ) y(kT ; 0, h, u) ˆ for all s ≥ kT. ≤ Cρ¯ e ∞ (u) ˆ < ∞. So Problem (L Q)∞ From this, one can easily verify that J0,h 0,h satisfies the FCC at h.

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