We give a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on the construction of an explicit strict Lyapunov function. We compare our sufficient condition with other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems. 1461 the boundary condition (1.4) is dissipative, i.e., implies that the equilibrium solution u ≡ 0 of system (1.1) with

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... e boundary condition (1.4) is exponentially stable. This problem has been considered in the literature for more than 20 years. To our knowledge, the first results were published by Slemrod in [21] and Greenberg and Li in [9] for the special case of 2 × 2 (i.e., u ∈ R 2 ) systems. A generalization to n × n systems was given by the Li school. Let us mention in particular [17] by Qin, [25] by Zhao, and [14, Theorem 1.3, page 173] by Li. All these results rely on a systematic use of direct estimates of the solutions and their derivatives along the characteristic curves. They give rise to sufficient dissipative boundary conditions which are kinds of "small gain conditions." The weakest sufficient condition [14, Theorem 1.3, page 173] is formulated as follows: ρ(|G ′ (0)|) < 1, where ρ(A) denotes the spectral radius of A ∈M n,n (R) and |A| denotes the matrix whose elements are the absolute values of the elements of A ∈M n,n (R). In this paper we follow a different approach, which is based on a Lyapunov stability analysis. The special case of 2 × 2 systems and F (u) diagonal has recently been treated in our previous paper [6] . In the present paper, by using a more general strict Lyapunov function (see section 4), we get a new weaker dissipative boundary condition, stated as follows: denotes the usual 2-norm of matrices in M n,n (R) and D n,+ denotes the set of diagonal matrices whose elements on the diagonal are strictly positive. Moreover, our proof is rather elementary, and the existence of a strict Lyapunov function may be useful for studying robustness issues. Our paper is organized as follows. In section 2, after some mathematical preliminaries, a precise technical definition of our new dissipative boundary condition is followed by the statement of our exponential stability theorem. Section 3 is then devoted to a discussion of the optimality properties of our dissipative boundary condition and to a comparison of this condition with other stability criteria from the literature, namely the criterion [14, Theorem 1.3, p, 173] mentioned above and a stability criterion for linear hyperbolic systems due to Silkowski. The proof of our exponential stability theorem, including the Lyapunov stability analysis, is thoroughly given in section 4. The paper ends with two appendices, where some technical properties of our dissipative boundary condition are given. A sufficient condition for exponential stability. For x := (x 1 ,...,x n ) tr ∈ C n , |x| denotes the usual Hermitian norm of x: For n ∈ N \{0} and m ∈ N \{0}, we denote by M n,m (R) the set of n × m real matrices. We define, for K ∈M n,m (R), K := max{|Kx|; x ∈ R n , |x| =1} = max{|Kx|; x ∈ C n , |x| =1}, and, if n = m, ρ 1 (K):=Inf{ ΔKΔ −1 ;Δ∈D n,+ }, (2.1)