Lp-Stability of a Class of Volterra Systems

This paper presents some new and explicit stability results for Volterra systems from two diﬀerent approaches. The ﬁrst approach is based on monomial domination of the Volterra system’s memoryless output nonlinearity and the second on its Lipschitz-norm. The former yields more widely applicable results, but introduces nonconvexity in the signal spaces for certain parameter values.

II. L p -STABILITY OF VOLTERRA SYSTEMS This paper focuses on the class of fading-memory Volterra systems, which includes a vast range of engineering applications.Furthermore, their specific structure allows concise analytical expressions to be obtained.
First, three distinct cases are considered: a monomial nonlinearity, a general monomially dominated nonlinearity and a Lipschitz-continuous nonlinearity.After a discussion of the results obtained, three more general cases are presented to pave the way toward analysis of multiple-input, multipleconvolution and multiple-output Volterra systems.
Case 1: Consider the following single-input single-convolution Volterra system: a Linear Time-Invariant (LTI) system with input signal x and impulse response g ∈ L 1 [0, ∞) that feeds into a memoryless nonlinearity of the form (•) v , for a given 0 < v < ∞.Its output is described by where * represents convolution.Even though the power function is memoryless, the output y possesses memory with respect to x due to the convolution featuring in (1).Young's convolution inequality [8, Th. 9.1] states that for all 1 ≤ p ≤ ∞, with • p the standard L p -norm on [0, ∞).
Using inequality (2), one obtains for all 1 ≤ vq ≤ ∞, or alternatively, 1/v ≤ q ≤ ∞.It follows that the Volterra system, described by (1), is finite-gain inputoutput stable from L vq [0, ∞) to L q [0, ∞) for every v > 0 and every q satisfying 1/v ≤ q ≤ ∞, both fixed.It is interesting to note that, for v > 1, the lower bound of q, namely 1/v, is less than one.However, for 1/v < q < 1, the L q -norm's triangle inequality is violated, causing every ε-ball in the output signal space L q [0, ∞) to be nonconvex [9], [10].On the other hand, if q is taken to be greater than or equal to one, then, for sufficiently small 0 < v < 1, it follows that vq < 1 so that every ε-ball in the input signal space L vq [0, ∞) is nonconvex.
The consequences of a Lebesgue exponent 1 strictly between zero and one are far reaching concerning the operator theory applicable to this Lebesgue space (see [11]).For example, any linear functional on this Lebesgue space is identically zero [12], implying that only nonlinear functionals on this space are worth studying.
Case 2: Consider a more general single-input singleconvolution Volterra system, with a memoryless output nonlinearity described by a bounded and measurable real-valued function h(•) that is dominated by the monomial (•) v , that is, This implies that h(0) = 0 and so it follows immediately that for all 1 ≤ vq ≤ ∞, based on the results derived for Case 1 above.Interestingly, in this last inequality, the norm of the nonlinear function h(•) does not appear on the right-hand side.From (3), it follows that the single-input single-convolution Volterra system with memoryless output nonlinearity being any measurable function h(•), dominated by the power-v function, is finite-gain input-output stable from L vq [0, ∞) to L q [0, ∞) for every v > 0 and every 1/v ≤ q ≤ ∞, both fixed.
Note that the above remarks about the nonconvexity of ε-balls in the Lebesgue spaces also apply here.
Case 3: To further extend the previous results, assume h(•) to be a Lipschitz-continuous memoryless nonlinearity.Consider the Lipschitz-norm defined by for fixed 1 ≤ p, q ≤ ∞, where • ∞ denotes the standard sup-norm [13], [14].Here, the subscript Lip(p, q) is introduced merely to explicitly indicate this norm's dependency on the domain and range spaces of h(•).
For every constant function h(•), the first term in • Lip(p,q) satisfies h ∞ = |h(0)| while the second term there, i.e. the slope of h(•), is identically zero. 2 This yields the following expression for the Lipschitz-norm: For the purpose of the discussion that follows, it is noted that for arbitrary but fixed arguments u 1 and u 2 on the right-hand side of the inequality, yielding 2 The quotient, in the expression for the Lipschitz-norm h Lip(p,q) of h(•), is a seminorm, which vanishes for every constant function h(•).
Since the intermediate signals, u 1 and u 2 , are the result of convolutions, the above inequality can be expressed as for some input signals x 1 and x 2 , after applying (2).
For the case of h(0) = 0, with h(•) not necessarily power-v dominated, by setting x 2 = 0, one has for all 1 ≤ p, q ≤ ∞.Therefore, for arbitrary but fixed 1 ≤ p, q ≤ ∞, a single-input single-convolution Volterra system with a memoryless output nonlinearity h(•) that has a finite Lipschitz-norm • Lip(p,q) and h(0) = 0, but is not necessarily dominated by the power-v function, is finite-gain input-output stable from L p [0, ∞) to L q [0, ∞).At this juncture, a caution is not to underestimate the role of the assumption h(0) = 0. To see this, note that assuming h(0) = 0 and x 2 ≡ 0 in (5) gives where 1(t) := 1 for all t ∈ [0, ∞).If one now attempts to apply the norm inequality z 1 − z 2 ≤ z 1 − z 2 , derived from the triangle inequality, to the left-hand side, then This is problematic because, for the standard L q [0, ∞) space, when q = ∞, one would have that 1(•) q = ∞ because 1(•) is not in the space L q [0, ∞).Therefore, one cannot isolate the term h(g * x 1 ) q by combining ( 5), ( 7) and ( 8) in the hope of deriving an upper bound for it.However, for the case of q = ∞, one has 1(•) ∞ = 1 and, therefore, one can now isolate the term h(g * x 1 ) ∞ by combining ( 5), ( 7) and ( 8), so as to obtain Consequently, the Volterra system considered here is finitegain input-output stable, even for h(0) = 0. Now, returning to (6), with h(0) = 0 therein, and setting p = q, one obtains h(g * x) q ≤ h Lip(q,q) g 1 x q , for an arbitrary input signal x ∈ L q [0, ∞) with 1 ≤ q ≤ ∞.
Comparing ( 9) with (3), it can be seen that the root (or powers) of v has been eliminated by including the factor consisting of the Lipschitz-norm of h(•) on the right-hand side.It is important to note that ( 9) and (3) give two alternative but not equivalent criteria for assessing the L p -stability of a Volterra system.In fact, (3) does not require h(•) to be Lipschitz-continuous; it even applies to the extreme case of h(•) being L q -integrable but nowhere-differentiable.
Case 4: Consider a two-input two-convolution Volterra system described by the equation where the real-valued memoryless nonlinearity h(•, •) is assumed to be Lipschitz-continuous and g 1 , g 2 ∈ L 1 [0, ∞).The Lipschitz-norm considered here is for fixed 1 ≤ p 1 , p 2 , q ≤ ∞, and the direct-sum norm 3  • p1,p2 , taken to be For arbitrary direct-sum signals (u 1 , s 1 ) and (u 2 , s 2 ), one has Substituting into this inequality then yields where the linearity of convolution and inequality (2) were used.Now, assuming that h(0, 0) = 0 holds and then setting x 12 = x 22 ≡ 0, finally gives h(g 1 * x 1 , g 2 * x 2 ) q ≤ h Lip(p1,p2,q) g 1 1 x 1 p1 + h Lip(p1,p2,q) g 2 1 x 2 p2 , (10) for arbitrary input signals ∞, fixed) and arbitrary 1 ≤ q ≤ ∞, fixed.Thus, if the Lipschitz-norm of h(•) is finite, then this Volterra system is finite-gain input-output stable from the direct-sum space Case 5: If the inputs are joined by setting x 2 = x 1 in Case 4, then one obtains a single-input two-convolution Volterra system.From (10), it then follows that where, in general, p 1 = p 2 .Under the assumption that h(•) has a finite Lipschitz-norm, the resulting Volterra system is finite-gain input-output stable from and this Volterra system is finite-gain input-output stable from L p [0, ∞) to L q [0, ∞). 3 There are many other direct-sum norms to choose from.Now, imposing power-v domination extends the earlier results.
Case 6: The last two more general cases easily generalize to a Volterra system with an arbitrary number of inputs, an arbitrary number of convolutions associated with each input, and an arbitrary number of outputs for both the cases of power-v domination and of Lipschitz-continuity of the output nonlinear memoryless mapping.
Finally, note that the exact same arguments presented above apply to the discrete-time case, thus producing equivalent results.

III. CONCLUSION
This paper presented some new and explicit stability results for Volterra systems when the output nonlinearity possesses a Lipschitz-norm as well as for the more general case of a measurable, monomially dominated output nonlinearity.These two approaches yield alternative but not equivalent stability criteria, thus providing more versatility by allowing selection of the more appropriate of the two for a given application.