3.1.2 A general uniqueness theorem The key issue for uniqueness of solutions turns out to be the maximal slope of a=a(a) to guarantee uniqueness on time interval T=[to, t,, it is sufficient to require existence of a constant M such that a(x1)-a(22)≤M|z1-2 for all 1, i2 from a neigborhood of a solution r: [ to, t]+R of (3.1). The proof of both existence and uniqueness is so simple in this case that we will formulate the statement for a much more general class of integral equations Theorem 3.1 Let X be a subset of r containing a ball B,(x)={∈R of radius r>0, and let ti> to be real numbers. Assume that function a: Xx[to, t, x to, ti kr is such that there exist constants M, K satisfying a(x1,7,t)-a(x2,T,圳≤K1-2V1,2∈B1(5o),to≤T≤t≤t1,(3.2) a(z,T,)≤MW∈B1(50),to≤r≤t≤t1 (3.3) for a sufficiently small t, to, there exists unique function a: [to, tfl satisfying r(t)=o+a(a(r),T, t)dr vtE[o, tyl (3.4) A proof of the theorem is given in the next section. When a does not depend on the third argument. we have the standard Ode case i(t)=a(a(t), t) In general, Theorem 3. 1 covers a variety of nonlinear systems with an infinite dimensional state space, such as feedback interconnections of convolution operators and memoryless which the forward loop is an LTI system with input u, output w, and transfer function e nonlinear transformations. For example, to prove well-posedness of a feedback system G(s and the feedback loop is defined by v(t)=sin(w(t), one can apply Theorem 3.1 with a(, T, t) sin(2)+h(t),t-1≤r≤t, h(t) otherwise where h=h(t)is a given continuous function depending on the initial conditions� 2 3.1.2 A general uniqueness theorem The key issue for uniqueness of solutions turns out to be the maximal slope of a = a(x): to guarantee uniqueness on time interval T = [t0, tf ], it is sufficient to require existence of a constant M such that |a(¯x1) − a(¯ ¯ x2)| ∃ M|x1 − x¯2| for all x¯1, x¯2 from a neigborhood of a solution x : [t0, tf ] ∈� Rn of (3.1). The proof of both existence and uniqueness is so simple in this case that we will formulate the statement for a much more general class of integral equations. Theorem 3.1 Let X be a subset of Rn containing a ball Br(¯x0) = {x¯ ≤ R x − ¯ n : |¯ x0| ∃ r} of radius r > 0, and let t1 > t0 be real numbers. Assume that function a : X × [t0, t1] × [t0, t1] ∈� Rn is such that there exist constants M, K satisfying |a(¯x1, �, t) − a(¯x2, �, t)| ∃ K|x¯1 − x¯2| � x¯1, x¯2 ≤ Br(¯x0), t0 ∃ � ∃ t ∃ t1, (3.2) and |a(¯x, �, t)| ∃ M � x¯ ≤ Br(¯x0), t0 ∃ � ∃ t ∃ t1. (3.3) Then, for a sufficiently small tf > t0, there exists unique function x : [t0, tf ] ∈� X satisfying t x(t) = x¯0 + a(x(� ), �, t)d� � t ≤ [t0, tf ]. (3.4) t0 A proof of the theorem is given in the next section. When a does not depend on the third argument, we have the standard ODE case x˙ (t) = a(x(t), t). In general, Theorem 3.1 covers a variety of nonlinear systems with an infinite dimensional state space, such as feedback interconnections of convolution operators and memoryless nonlinear transformations. For example, to prove well-posedness of a feedback system in which the forward loop is an LTI system with input v, output w, and transfer function e−s − 1 G(s) = , s and the feedback loop is defined by v(t) = sin(w(t)), one can apply Theorem 3.1 with sin(¯x) + h(t), t − 1 ∃ � ∃ t, a(¯x, �, t) = h(t), otherwise, where h = h(t) is a given continuous function depending on the initial conditions