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Variation of constants for a finite-matrix action defect

Statement

Let $n \geq 1$ , $A \in M_{n} (C)$ , $T > 0$ , $v \in C^{n}$ , and $u : [- T, T] \to C^{n}$ be continuously differentiable, with one-sided derivatives at the endpoints. Define the continuous defect $r (s) : = A u (s) - u^{'} (s)$ . Then, for every $t \in [- T, T]$ ,

$e^{t A} v - u (t) = e^{t A} (v - u (0)) + \int_{0}^{t} e^{(t - s) A} r (s) d s .$

The vector integral is the oriented componentwise Riemann integral after identifying $C^{n}$ with $R^{2 n}$ . In particular, if $u (0) = v$ , the initial term vanishes.

Facts & Assumptions

Given: $n \geq 1$ , a complex square matrix $A$ , $T > 0$ , a vector $v$ , and a continuously differentiable curve $u : [- T, T] \to C^{n}$ with defect $r (s) = A u (s) - u^{'} (s)$ .

[F1]

The exponential series of a bounded operator: for a bounded operator $A$ , $E (t) : = e^{t A}$ is differentiable, $E^{'} (t) = A E (t) = E (t) A$ , and $E (t) E (s) = E (t + s)$ .

[F2]

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral: derivatives and integrals of finite-dimensional real vector-valued curves are taken componentwise.

[F3]

If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz: if a real vector-valued curve is differentiable on a compact interval and its derivative is integrable, the integral of the derivative equals the endpoint difference.

[F4]

$C$ is the real coordinate plane, with coordinate arithmetic and The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products: real and imaginary coordinates identify $C^{n}$ with a finite-dimensional real vector space, and the standard inner product is given coordinatewise.

[F5]

Every finite-dimensional normed space is Banach: the finite-dimensional complex-inner-product space $C^{n}$ is complete.

[F6]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: for a linear map between finite-dimensional inner-product spaces, its operator norm is the maximum of $∥ A x ∥_{2}$ over unit vectors.

[F7]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: every continuous real-valued function on a nondegenerate closed bounded interval is Riemann integrable.

Proof

technique · direct

1.1givenF1F5F6algebra

Set $x (t) : = e^{t A} v - u (t)$ . If $y \neq 0$ , rescaling $y / ∥ y ∥_{2}$ in [F6] gives $∥ A y ∥_{2} \leq ∥ A ∥_{2} ∥ y ∥_{2}$ , while the same inequality is immediate for $y = 0$ ; thus $A$ is bounded. By [F1] and [F5], the exponential series applies on the complete space $C^{n}$ . It gives $x^{'} (t) = A e^{t A} v - u^{'} (t) = A (x (t) + u (t)) - u^{'} (t) = A x (t) + r (t)$ , and $x (0) = v - u (0)$ .

2.1step 1.1F1algebra

Define $g (t) : = e^{- t A} x (t)$ . Differentiating the product and using that $A$ commutes with every power in its exponential series gives $g^{'} (t) = - A e^{- t A} x (t) + e^{- t A} (A x (t) + r (t)) = e^{- t A} r (t)$ .

3.1step 2.1F2F3F4F7

By [F2] and [F4], $g$ is a real finite-dimensional vector-valued curve with continuous derivative. Each coordinate of $g^{'}$ is continuous and therefore Riemann integrable on every nondegenerate closed subinterval by [F7]. If $t = 0$ , then $g (t) = g (0)$ and the oriented integral is zero. If $t > 0$ , [F3] on $[0, t]$ gives $g (t) = g (0) + \int_{0}^{t} e^{- s A} r (s) d s$ . If $t < 0$ , [F3] on $[t, 0]$ gives $g (0) - g (t) = \int_{t}^{0} e^{- s A} r (s) d s$ , which is the same oriented identity after reversing the bounds.

4.1step 3.1F1F2algebra∎

Multiply this equality by $e^{t A}$ . Matrix multiplication is a fixed finite-dimensional linear map and therefore commutes with the componentwise integral; [F1] gives $e^{t A} g (t) = x (t)$ , $e^{t A} g (0) = e^{t A} (v - u (0))$ , and $e^{t A} e^{- s A} = e^{(t - s) A}$ . Substitution yields the stated identity.

Depends on

Used by

Arnoldi action error is the semigroup-weighted residual integral Theorem

Dependency tree · two levels

86 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Awad H. Al-Mohy and Nicholas J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators (standard reference, not scraped)
Nicholas J. Higham and Lijing Lin, Functions of Matrices: A Short Course (standard reference, not scraped)