By Peter J. Olver

ISBN-10: 0387940073

ISBN-13: 9780387940076

ISBN-10: 3540940073

ISBN-13: 9783540940074

Symmetry tools have lengthy been famous to be of significant value for the examine of the differential equations. This booklet offers a pretty good advent to these functions of Lie teams to differential equations that have proved to be beneficial in perform. The computational tools are offered in order that graduate scholars and researchers can simply discover ways to use them. Following an exposition of the functions, the ebook develops the underlying concept. some of the themes are provided in a unique means, with an emphasis on particular examples and computations. extra examples, in addition to new theoretical advancements, look within the workouts on the finish of every bankruptcy.

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**Additional info for Applications of Lie Groups to Differential Equations**

**Example text**

Let n U(k) = u : ui ≥ 0, i = 1, . . , n, ui ≤ k i=1 denote the set of feasible controls given the machine capacity k. Let A(k) denote the collection of all admissible controls with the initial condition k(0) = k. 1. 6). The extension of results to other cases is standard. 2. A function u(·, ·) deﬁned on n ×M is called an admissible feedback control, or simply a feedback control, if: (i) for any given initial surplus x and production capacity k, the equation d x(t) = u(x(t), k(t)) − z dt has a unique solution; and (ii) the control deﬁned by u(·) = {u(t) = u(x(t), k(t)), t ≥ 0} ∈ A(k).

45) 44 3. Optimal Control of Parallel-Machine Systems which contradicts the deﬁnition of V ρ (x, k). Thus (ii) is proved. 1. 30) is also a potential function. , λ = λ∗ . 2. In the simple special case (m = 1, c(u) = 0, h(x) = h1 x+ + h2 x− ) solved by Bielecki and Kumar [20], we note that optimality is shown only over the class of stable controls and not over the (natural) class of admissible controls. That is, they prove (i) and (iii) but not (ii). 2, we know that the relative cost function V (x, k) ∈ G.

1 .. λm−1 .. −(λm−1 + µm−1 ) µm−1 λm −λm with µj > 0 (0 ≤ j ≤ m − 1) and λj > 0 (1 ≤ j ≤ m), then yi0 ≥ yi0 +1 ≥ · · · ≥ ym . ⎟ ⎟ ⎟ ⎟ ⎠ 52 3. Optimal Control of Parallel-Machine Systems Proof. Suppose contrariwise that ym−1 < ym . First we show that from this assumption, ym−2 < ym−1 . 1, we have 0= dh(ym ) + λm dx ∂V (ym , m − 1) ∂V (ym , m) − ∂x ∂x . 66) and ∂V (ym−1 , m − 1) ∂V (ym , m) ∂V (ym , m − 1) ∂V (ym , m) − > − . 54), we have ∂V (ym−1 , m − 1) ∂V (ym , m) dc(z) dc(z) − =− + = 0.

### Applications of Lie Groups to Differential Equations by Peter J. Olver

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