# Foundations of modern analysis (His Treatise on analysis, by Jean Alexandre Dieudonne

By Jean Alexandre Dieudonne

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This ebook includes lecture notes of a summer time institution named after the overdue Jacques Louis Lions. The summer time tuition was once designed to alert either Academia and to the expanding function of multidisciplinary tools and instruments for the layout of complicated items in a variety of parts of socio-economic curiosity.

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3 Stabilization 37 and ∀s ∈ {1, 2, . . 5) such that the closed-loop system is stochastically stable. 26). 19) holds. 19) is equivalent to ⎡ −Pi ∗ ∗ ⎢ √πiK1 PK1 Ai −PK1 ∗ ⎢ √ ⎢ πiK2 PK2 Ai 0 −PK2 ⎢ ⎢ .. .. ⎢ . . ⎢√ ⎢ πiKm PKm Ai 0 0 i i ⎣ (i) 1 − πK Pj Ai 0 0 ∗ ∗ ∗ .. 40). 37). 14, that is, the underlying system is stochastically stable. 26). 22 In contrast with the continuous-time case, the discrete-time case is relatively simpler since all the elements in the TPM are nonnegative and we need not distinguish the cases of diagonal elements known or unknown.

10), one can also readily obtain Θi < 0. 3), which completes the proof. 8 Note that if IU(i)K = ∅, ∀i ∈ I, the underlying system is the one with completely known TRs, which becomes the MJLS in the usual sense. 6). , the TRs are completely unknown, then the system can be if IK viewed as a switching linear system under arbitrary switching. 10) becomes Ai Pi + Pi Ai ≤ −Pj , ∀i = j ∈ I. , Pi = P. 10) are reduced to Ai P + PAi = −P < 0, namely, a latent quadratic common Lyapunov function will be shared among all the modes.

0, ⎥ . √ (i) (i) ⎥ πiKm PKm Ai ⎦ (i) −πK Pi 1 1 −Pj Pj Ai ∗ −Pi < 0. 36), respectively. 31). 32). e. the underlying system is stochastically stable. 26). This completes the proof. From the development in the above theorems, one can clearly see that our obtained stability and stabilization conditions actually cover the results for the usual MJLS and the switching linear systems under arbitrary switching (all the TPs are unknown). Therefore, the systems considered and corresponding criteria explored in the section are more general in hybrid systems field.