ELEC4631 Chap.4 Lyapunov Stability I: Direct Method, Energy Functions and Invariant Sets
Lyapunov Stability I: Direct Method, Energy Functions and Invariant Sets
Lyapunov’s direct method replaces an explicit trajectory solution with a scalar certificate. Begin by finding the equilibrium and, if necessary, shifting coordinates. A candidate V must be positive about that equilibrium on a declared domain; its derivative is computed along the vector field as grad(V)^T f(x). Positive V with nonpositive derivative supports stability, while a negative-definite derivative gives asymptotic stability. Global claims additionally need a global domain and typically radial unboundedness so level sets do not escape. When the derivative is only negative semidefinite, LaSalle’s invariance principle asks for the largest invariant subset of the zero-derivative set; the set itself is not the answer. This chapter’s course-specific answer shape lists positivity, derivative sign, invariant-set or radial facts and then states the precise local/global stability conclusion.
What this chapter covers
- 01Equilibria and coordinate shifts
- 02Lyapunov stability definitions
- 03Positive definite and radially unbounded candidates
- 04Derivative along trajectories
- 05Direct-method theorem conclusions
- 06LaSalle largest invariant subset
- 07Candidate construction and cross-term cancellation
Global certificate with cross-term cancellation
- equilibriumThe origin is an equilibrium because both right-hand sides vanish at x=0.
- positivityV(0)=0 and V(x)>0 for every nonzero x. V is positive definite globally.
- radialV tends to infinity with the state norm, so it is radially unbounded.
- derivativeV-dot=x1(-x1+x2)+x2(-x1-x2^3)=-x1^2-x2^4; cross-terms cancel.
- signThe derivative is negative definite because it is zero only at the origin.
- scopeAll properties hold globally, so the origin is globally asymptotically stable.
Key terms
- Equilibrium
- A point x* satisfying f(x*)=0. Stability is defined about an equilibrium, so this check precedes every certificate.
- Positive definite function
- V(x*)=0 and V(x)>0 for all other points in the stated neighbourhood or domain.
- Radially unbounded
- V(x) tends to infinity as the state norm tends to infinity, helping extend bounded level-set arguments globally.
- Derivative along trajectories
- V-dot=grad(V)^T f(x), not a free partial derivative with respect to a visible time variable.
- Invariant set
- A set a trajectory cannot leave. LaSalle uses the largest invariant subset contained in {x:V-dot=0}.
- Region of attraction
- The set of initial states whose trajectories converge to the equilibrium. A local certificate proves only an inner estimate unless more is shown.
Lyapunov Stability I: Direct Method, Energy Functions and Invariant Sets FAQ
Does V-dot<=0 prove asymptotic stability?
Not by itself. It usually proves stability under the other direct-method conditions. To establish convergence with a semidefinite derivative, analyse the largest invariant subset of the zero-derivative set, often through LaSalle. If that invariant subset contains only the equilibrium, asymptotic stability can follow.
When may I say globally asymptotically stable?
The candidate and derivative conditions must hold over the whole state space or declared global domain, and level sets must prevent escape—commonly through radial unboundedness. A local polynomial inequality near the origin does not become global simply because the algebra is short.
How do I invent a Lyapunov candidate?
Start with physically meaningful energy or a positive quadratic form. Differentiate it and inspect troublesome cross-terms. Modify coefficients so terms cancel or dominate them with inequalities. Candidate construction is guided iteration: positivity and derivative sign must be checked after every adjustment.
What if the equilibrium is not the origin?
Solve f(x*)=0, define a deviation z=x-x* and express the dynamics and candidate about z=0. Writing V(x)>0 about the origin when the vector field does not vanish there cannot certify stability of the intended operating point.
What does a theorem-complete Lyapunov answer look like?
First solve f(x*)=0 and declare the domain. State V(x*)=0 and justify positive definiteness—by completing squares, eigenvalues of a quadratic matrix or direct inequalities. Compute the gradient and dot it with the full vector field, showing cancellations and sign-sensitive steps. Classify V-dot as negative definite, negative semidefinite or sign-indefinite. If semidefinite, identify E={x:V-dot=0}, test the dynamics on E and find its largest invariant subset. For a global claim, explain why the properties hold globally and why level sets are bounded, commonly through radial unboundedness. Finish with exactly one supported adjective. This sequence is longer than “V positive, derivative negative,” but each clause answers a distinct theorem condition and makes partial working assessable.
How should I test or improve a candidate function?
Start with the simplest positive quadratic or physical energy. Differentiate it before adding complexity. If cross terms prevent a sign conclusion, change the relative coefficients, add a cross term while preserving positive definiteness, or dominate the cross term with a bound such as 2ab<=epsilon a^2+(1/epsilon)b^2. Recheck positivity after every modification; a candidate tailored only to make the derivative attractive may cease to be positive. Plotting or sampling can reveal a counterexample but does not prove positivity over a continuum. Use computation to search for failures, then present the analytical inequality or eigenvalue argument. If no candidate works, the failure is not a proof of instability—it may only mean the chosen family is too restrictive.
What practice sequence makes LaSalle less mechanical?
Take one system whose V-dot is negative definite, one whose zero-derivative set is a line but only the origin is invariant, and one whose zero set contains a genuine orbit or continuum of equilibria. For each, write E={V-dot=0}, substitute its defining conditions into the vector field and ask whether trajectories remain there. Predict the conclusion before invoking the theorem. Use plots only to build intuition, then supply the invariant-set algebra. If an AI tutor generates variants, request the equilibrium and candidate but hide its solution until you have identified E and the largest invariant subset yourself.
Exam move
Use a seven-line proof template: equilibrium, domain, V(0), V positivity, V-dot calculation, derivative classification/invariant set, conclusion with scope. Practise one example with negative-definite derivative and one with a semidefinite derivative whose zero set is larger than the invariant subset. Sketch level sets and vector-field directions to make invariance concrete. When a candidate includes cross-terms, test symmetry and positive definiteness before differentiation. After obtaining V-dot, resist the urge to write the adjective immediately; ask whether strictness, radial unboundedness and domain support local stability, local asymptotic stability or a global result. The older reference-answer structure reinforces this property-list-first habit without supplying current assessment numbers.
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