On complex-valued 2D eikonals. Part four: continuation past a caustic

Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -- in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension $2.$ We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions.Comment: 48 pages, 15 figure

A Theory of Strongly Continuous Semigroups in Terms of Lie Generators

AbstractLetXdenote a complete separable metric space, and let C(X) denote the linear space of all bounded continuous real-valued functions onX. A semigroupTof transformations fromXintoXis said to be jointly continuous if the mapping (t, x)→T(t)xis jointly continuous from [0, ∞)×XintoX. The Lie generator of such a semigroupTis the linear operator in C(X) consisting of all ordered pairs (f, g) such thatf, g∈C(X), and for eachx∈X,g(x) is the derivative at 0 off(T(·)x). We completely characterize such Lie generators and establish the canonical exponential formula for the original semigroup in terms of powers of resolvents of its Lie generator. The only topological notions needed in the characterization are two notions of sequential convergence, pointwise and strict. A sequence in C(X) converges strictly if the sequence is uniformly bounded in the supremum norm and converges uniformly on compact subsets ofX

Nonlinear systems with unbounded controls and state constraints: a problem of proper extension

The authors consider the control system described by x\u2d9 = f(t, x, u, v, u\u2d9 ), t 2 (t, T], with initial data (x, u)(t) = (x, u), where f is sublinear in u\u2d9 , and the state variable is constrained in the closure of an open set Rm; u and v take values in a closed subset U Rm and a compact set V Rq, respectively. Here, v is a bounded, measurable control, while no bounds are imposed on u\u2d9 . By introducing a space-time vector field f defined by f(t, x, u, v,w0,w) = f(t, x, u, v,w/w0)w0 if w0 6= 0, f1(t, x, u, v,w) if w0 = 0, where f1(t, x, u, v,w) def = limr!1 r 121f(t, x, u, v, rw), the original system is embedded in the extended system t0 = w0(s), x0 = f(t, x, u(s), v(s),w0(s),w(s)), u0 = w(s), (t, x, u)(0) = (t, u, u), s 2 [0, 1]. The main result consists of conditions which guarantee that each constrained trajectory of the extended system is the limit of trajectories of the original system