Consider the differential equation 𝐮̇=-K 𝐮, where K is positive semi-definite. (a) Find all equi

step 1 In the problem, we have DQ upon DT equals Q upon K or integration DQ upon Q equals 1 upon K DT.

Consider the differential equation 𝐮̇=-K 𝐮, where K is...

Consider the differential equation $\dot{\mathbf{u}}=-K \mathbf{u}$, where $K$ is positive semi-definite. (a) Find all equilibrium solutions. (b) Prove that all non-constant solutions decay exponentially fast to some equilibrium. What is the decay rate? (c) Is the origin stable, asymptotically stable, or unstable? (d) Prove that, as $t \rightarrow \infty$, the solution $\mathbf{u}(t)$ converges to the orthogonal projection of its initial vector $\mathbf{a}=\mathbf{u}(0)$ onto ker $K$.

Consider the differential equation $\dot{\mathbf{u}}=-K \mathbf{u}$, where $K$ is positive semi-definite.
(a) Find all equilibrium solutions. (b) Prove that all non-constant solutions decay exponentially fast to some equilibrium. What is the decay rate? (c) Is the origin stable, asymptotically stable, or unstable? (d) Prove that, as $t \rightarrow \infty$, the solution $\mathbf{u}(t)$ converges to the orthogonal projection of its initial vector $\mathbf{a}=\mathbf{u}(0)$ onto ker $K$.

Key Concepts

Equilibrium Solutions

An equilibrium solution of a differential equation is a solution that remains constant in time. In the context of a linear differential equation with a matrix operator, equilibrium solutions are precisely those vectors that are annihilated by the matrix; that is, they reside in the kernel (null space) of the matrix operator. This concept is fundamental in linear dynamical systems where the operator's properties dictate the system's long?term behavior.

Exponential Decay

Exponential decay in a linear system indicates that solutions decrease in norm at a rate governed by the eigenvalues of the system's matrix outside of any zero eigenvalues. In systems with a positive semi-definite operator, all components associated with non-zero eigenvalues decay exponentially. The decay rate is typically determined by the smallest positive eigenvalue of the operator, which facilitates an understanding of how fast perturbations away from equilibrium diminish.

Stability

Stability in the context of linear differential equations refers to whether small deviations from an equilibrium result in motions that remain nearby or diverge over time. A system is stable if trajectories starting close to an equilibrium remain bounded, and it is asymptotically stable if, in addition, they converge to the equilibrium as time tends to infinity. However, if the operator has a non-trivial kernel, there may be a continuum of equilibrium points, affecting the classification of stability of specific equilibria such as the origin.

Orthogonal Projection

The concept of orthogonal projection onto a subspace is fundamental in linear algebra and appears in the analysis of the long-time behavior of linear systems. When the system dynamics annihilate components in certain subspaces (such as the image of the operator) and leave others intact (such as the kernel), the solution converges to the projection of its initial condition onto the invariant subspace (in this case, the kernel). This principle is critical for understanding how, despite the decay of some components, the remaining components determine the system's asymptotic state.

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