Strict Lyapunov Function Constructions under LaSalle Conditions with an
Application to Lotka-Volterra Systems
Mathematical control theory provides the theoretical foundations that
undergird many modern technologies, including aeronautics, biotechnology,
communications networks, manufacturing, and models of climate change. During
the past fifteen years, there have been numerous exciting developments at
the interface of control engineering and mathematical control theory. Many
of these advances were based on new Lyapunov methods for analyzing and
stabilizing nonlinear systems. Constructing strict Lyapunov functions is a
central and challenging problem. On the other hand, non-strict Lyapunov
functions are often constructed easily, using passivity, backstepping, or
forwarding, or by taking the Hamiltonian for Euler-Lagrange systems. Roughly
speaking, non-strict Lyapunov functions are characterized by having negative
semi-definite time derivatives along all trajectories of the system, while
strict Lyapunov functions have negative definite derivatives along the
trajectories. Even when we know a system to be globally asymptotically
stable, it is often still important to have an explicit global strict
Lyapunov function, e.g., to design feedbacks that give input-to-state
stability to actuator errors.
One important research topic involves finding necessary and sufficient
conditions for different kinds of stability, in terms of the existence of
Lyapunov functions, such as Lyapunov characterizations for hybrid systems,
or for systems with measurement uncertainty and outputs. Some of the most
significant recent work in this direction has been carried out by Andrew
Teel and his co-workers, who employ systems on hybrid time domains that
encompass continuous time and discrete time systems as special cases.
Converse Lyapunov function theory implies the existence of strict Lyapunov
functions for large classes of globally asymptotically stable nonlinear
systems. However, the Lyapunov functions given by converse theory are often
abstract or non-explicit, and so may not always lend themselves to feedback
design. Explicit strict Lyapunov functions are also important for
quantifying the effects of uncertainty, because, e.g., they can be used to
build the comparison functions in the input-to-state stability estimate. In
fact, once we construct a suitable global strict Lyapunov function, several
significant stabilization and robustness problems can be solved almost
immediately, using standard arguments.
In some situations, non-strict Lyapunov functions are enough, because they
can be used in conjunction with LaSalle Invariance or Barbalat's Lemma to
show global asymptotic stability. In other cases, it suffices to analyze the
system around a reference trajectory, or near an equilibrium point, so
linearizations having simple local quadratic Lyapunov functions apply.
However, it is now well appreciated that linearizations and non-strict
Lyapunov functions are insufficient to analyze general time-varying
nonlinear systems. Non-strict Lyapunov functions are not well suited for
robustness analysis, because their negative semi-definite time derivatives
along the trajectories could become positive under small uncertainties of
the system. Uncertainties usually arise in applications, because of unknown
model parameters, or noise entering controllers. For this reason,
input-to-state stability and other robustness proofs often rely on finding
global strict Lyapunov functions. Also, there are important classes of
nonlinear systems (such as chemostat models) that often evolve far from
their equilibria. This has motivated a significant body of research on ways
to explicitly construct strict Lyapunov functions.
One approach to designing explicit strict Lyapunov functions, which has
received a lot of attention in the past few years, is the so-called
strictification approach. This entails transforming given non-strict
Lyapunov functions into explicit global strict Lyapunov functions. The
technique has been successfully employed in many contexts, including
adaptive control, rapidly and slowly time varying systems, and Hamiltonian
systems satisfying the conditions from the Jurdjevic-Quinn theorem.
Strictification reduces difficult strict Lyapunov function construction
problems to oftentimes much simpler non-strict Lyapunov function
construction problems. This talk presents two recent strictification
methods. The first relies on an appropriate nondegeneracy condition on the
higher order Lie derivatives of the nonstrict Lyapunov function in the
direction of the system dynamics, and the second gives a general procedure
for choosing the auxiliary functions in Matrosov's theorem. The simplicity
of our constructions makes them suitable for quantifying the effects of
uncertainty, and for feedback design. We illustrate our work using the
Lotka-Volterra model, which plays a fundamental role in bioengineering.
For related papers, see http://www.math.lsu.edu/~malisoff/research.html.