We propose a bundle trust-region algorithm to minimize locally Lipschitz functions which are potentially nonsmooth and nonconvex. We prove global convergence of our method and show by way of an example that the classical convergence argument in trust-region methods based on the Cauchy point fails in the nonsmooth setting. Our method is tested experimentally on three problems in automatic control.The question is then whether there are more restrictive classes of nonsmooth functions, where the Cauchy point can be salvaged. In response we show that the classical trust-region strategy with Cauchy point is still valid for upper C 1 -functions, and at least partially, for functions having a strict standard model. It turns out that several problems in control and in contact mechanics are in this class, which justifies the disquisition. Nonetheless, the class of functions where the Cauchy point works remains exceptional in the nonsmooth framework, which is corroborated by the fact that it does not include nonsmooth convex functions.A strong incentive for the present work comes indeed from applications in automatic control. In the experimental part we will apply our novel bundle trust-region method to compute locally optimal solutions to three NP-hard problems in the theory of systems with uncertain parameters. This includes (i) computing the worst-case H ∞ -norm of a system over a given uncertain parameter range, (ii) checking robust stability of an uncertain system over a given parameter range, and (iii) computing the distance to instability of a nominally stable system with uncertain parameters. In these applications the versatility of the bundle trust-region approach with regard to the choice of the norm is exploited.Nonsmooth trust-region methods which do not include the possibility of bundling are more common, see for instance Dennis et al. [17], where the authors present an axiomatic approach, and [13, Chap. 11], where that idea is further expanded. A recent trust-region method for DC-functions is [26].The structure of the paper is as follows. The algorithm is developed in section 2, and its global convergence is proved in section 3. Applications of the model approach are discussed in section 5, where we also discuss failure of the Cauchy point. Numerical experiments with three problems in automatic control are presented in section 6. NotationFor nonsmooth optimization we follow [12]. The Clarke directional derivative of f is

We analyze robust stability and performance of dynamical systems with real uncertain parameters. We compute criteria like the distance to instability, the worst-case spectral abscissa, or the worst-case H∞-norm, which quantify the degree of robustness of such a system when parameters vary in a given set ∆. As computing these indices is NP-hard, we present a heuristic which finds good lower bounds fast and reliably. Posterior certification is then obtained by an intelligent global strategy. A test bench of 87 systems with up to 70 states 39 uncertain parameters with up to 11 repetitions demonstrates the potential of our approach.1 Problem Specification.Robustness specifications limit the loss of performance and stability in a system where differences between the mathematical model and reality crop up. Robustness against real parametric uncertainties is among the most challenging calls in this regard. Already deciding whether a given system with uncertain real parameters δ is robustly stable over a given parameter range δ ∈ ∆ is NP-hard, and this is aggravated when it comes to deciding whether a given level of H 2 -or H ∞ -performance is guaranteed over that range. In this work we address this type of uncertainty by computing three key criteria, which quantify the degree of parametric robustness of a system. These are (a) the worst-case H ∞ -norm, and (b) the worst-case spectral abscissa over a given parameter range, and (c) the distance to instability, or stability margin, of a system with uncertain parameters.Consider a Linear Fractional Transform [23] with real parametric uncertainties as in Figure 1, whereand x ∈ R n is the state, w ∈ R m1 a vector of exogenous inputs, and z ∈ R p1 a vector of regulated outputs. The uncertainty channel is defined as p = ∆q, where the time-invariant uncertain matrix ∆ has the blockdiagonal form2) * ONERA, Control System Department, Toulouse, France † Université de Toulouse, Institut de Mathématiques with δ 1 , . . . , δ m representing real uncertain parameters, and r i giving the number of repetitions of δ i . Here we assume without loss that δ = 0 ∈ ∆ represents the nominal parameter value, and we consider δ ∈ ∆ in one-to-one correspondence with the matrix ∆ in (1.2). For practical applications it is generally sufficient to consider the caseTo analyze the performance of (1.1) in the presence of the uncertain δ ∈ R m we compute the worst-casewhere · ∞ is the H ∞ -norm, and where T wz (s, δ) is the transfer function z(s) = F u (P (s), ∆)w(s), obtained by closing the loop between (1.1) and p = ∆q with (1.2) in Figure 1. The solution δ * ∈ ∆ of (1.3) represents a worst possible choice of the parameters δ ∈ ∆, which may be an important element in analyzing performance and robustness of the system, see e.g. [1].Our second criterion is similar in nature, as it allows to verify whether the uncertain system (1.1) is robustly stable over a given parameter range ∆. This can be tested by maximizing the spectral abscissa of the system A-matrix over the parameter range−1 C q , and w...

We use a non-smooth trust-region method for H ∞ -control of infinite-dimensional systems. Our method applies in particular to distributed and boundary control of partial differential equations. It is computationally attractive as it avoids the use of system reduction or identification. For illustration the method is applied to control a reaction-convection-diffusion system, a Van de Vusse reactor, and to a cavity flow control problem. remained open in [5]. This given an affirmative answers to a question already posed in [26] for the convex non-smooth trust-region method. For complementary information on bundle methods see [27,15,22], a mix of bundle and trust-regions is [30].Design of controllers in the frequency domain based on non-smooth optimization has already been performed in [20,21,23,24,19,9]. Structured H − ∞-control for infinitedimensional systems is addressed in [3]. H ∞ -control of a heat exchange system is discussed in [29]. These approaches use either unstructured controllers, are based on matrix inequalities, or differ with regard to the optimization technique.The structure of the paper is as follows. In section 2 we outline our approach to H ∞ -control of infinite dimensional systems. In section 3 we discuss optimization and present our non-smooth trust-region method originally proposed in [5], on which the present approach rests. Convergence of the non-smooth trust-region method is discussed in section 3.2. In section 4 we point to some particularities when applying the trust-region method to H ∞ -optimization. This concerns the choice of working model, trial step, and stability barrier, as well as the approximation error between the infinite-dimensional H ∞program and its discretization.Numerical results for applications to infinite-dimensional control problems are presented in section 5. Subsection 5.1 shows how the method is, in general, applied to a boundary control problem, subsection 5.2 illustrates this in boundary H ∞ -control of a non-linear reaction-convection-diffusion equation, and subsection 5.3 for a non-linear Van de Vusse reactor. Subsection 5.4 discusses a cavity flow control problem.

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