By René Dáger

This quantity offers an in depth learn of partial differential equations on planar graphs modeling networked versatile mechanical constructions. exact emphasis is laid at the knowing of wave propagation phenomena, in the course of the research of the issues of observability and controllability from small areas of the graph or its boundary. a few of these effects are prolonged to the warmth, beam and Schrödinger equations on planar graphs. Designed as a self-contained introductory direction on regulate and commentary of networks, the amount includes additionally a few complicated themes and new suggestions that could be of curiosity for researchers during this sector. it is also a listing of open difficulties and themes for destiny examine.

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19) may be expressed in terms of the Fourier coefficients ¯ ,φ ¯ ) as (φ0,n ), (φ1,n ) of the initial data (φ 0 1 T r C 0 j=1 |∂n φi (t, vj )|2 dt ≥ µn φ20,n + φ21,n . 7. 19) is not true, that is, such that there ¯1 ) ∈ V ′ × H, which are not controllable in time T . 21) n∈N with non-vanishing coefficients cn . 22) is controllable in time T . From that fact, it would hold, in particular, that the system is spectrally controllable (and then approximately controllable) in time T . 22). This would imply that the space Wε is controllable in time T .

Then, (¯ u0 , u ¯1 ) ∈ WT if, and only if, ¯ , −φ ¯ ) (¯ u0 , u ¯1 ), (φ 1 0 (H×V ′ )×(H×V ) ¯ P∗ (φ ¯ , −φ ¯ ) = h, 1 0 T U ¯ ∂n φ| ¯C = h, U. 5. 21) with initial state (φ ¯ ,φ ¯ ). 6. 35) suggests a minimization algorithm for the ¯ If we look for the control in the form h ¯ = construction of the control h. 35) is the Euler equation I ′ (ψ 0 1 quadratic functional I : V × H → R defined by ¯ ,φ ¯ )= 1 I(φ 0 1 2 T 0 r j=1 ¯ − u ¯ . 35) will be verified. Therefore, if (ψ 0 1 The functional I is continuous and convex.

This allows, in particular, to prove observability properties of the solutions of the 1 − d wave equation when measurements are done only on one end-point of the string. 1 D’Alembert Formula Let us assume that the function u(t, x) satisfies the 1 − d wave equation in R × R. Then, for every t∗ ∈ R the function u may be expressed by means of the D’Alembert formula u(t, x) = 1 1 (u(t∗ , x + t − t∗ ) + u(t∗ , x − t + t∗ )) + 2 2 x+t−t∗ ut (t∗ , ξ)dξ. 1) is also valid if we change their role. Thus, if u(t, x) satisfies the 1 − d wave equation in R × [0, ℓ] then, for every a ∈ [0, ℓ], u(t, x) may be expressed by the sidewise formula u(t, x) = 1 1 (u(t + x − a, a) + u(t − x + a, a)) + 2 2 t+x−a ux (τ , a)dτ .