Convex In Direction Research Articles

On the maximum priciple for nonlinear discrete-time systems

It was shown in a previous paper that Halkin's maximum principle for difference equations linear in the state could be generalized by relaxing a convexity requirement to directional convexity. We show here that Halkin's results for difference equations not necessarily linear in the state may also be generalized. The new proof uses some rather simple arguments. The proof is based on a condition characterizing the optimal solution. This condition may be used to strengthen the maximum principle in the abnormal case.

Convexity and the maximum principle for discrete systems

Halkin has given a derivation of the discrete maximum principle using a convexity requirement. An example given in this paper shows that incorrect results may be obtained when Halkin's convexity requirement is not met. There are, however, systems that do not satisfy the convexity requirement, but for which there is still a maximum principle. The discrete maximum principle is rederived with a requirement, directional convexity, that is weaker than convexity and which considerably extends its applicability. Though convexity has appeared to be basic in the development of optimal control theory, it is only the weaker property of directional convexity which is required for much of the development.

On the Decomposition of a Choquet Simplex into a Direct Convex Sum of Complementary Faces.

On the Decomposition of a Choquet Simplex into a Direct Convex Sum of Complementary Faces.

Submerged stabilized platform : Alldredge L.R. and J.C. Fitz, 1964. Deep-Sea Res., 11 (6): 935dash942

Coupled structural and paleomagnetic analyses have shown that the northern Iberian bend of the Variscan orogen, referred to as the Cantabrian Orocline, developed by vertical axis rotation of an originally linear orogen. However, palinspastic restoration of the orocline has proven difficult owing to (1) an unusually great orogenic width of over 700 km and (2) exposure of shallow water strata of the Gondwanan margin in the northern and southern portions of the orogen. We present paleocurrent data from Lower Ordovician shallow marine clastic sedimentary rocks across the Variscan of northern and central Iberia collected to constrain palinspastic restoration of the orogen. Paleocurrent data were collected from over 50 sites, and include cross bed foresets, ripple crests and casts, as well as rare ball and pillow structures, syn-sedimentary slump folds, and incised channels. Paleocurrent directions fan around the Cantabrian Orocline, are consistently oriented at a high angle to structural strike, and yield a consistent offshore direction outward from the oroclinal core. Similarly, changes in structural strike and paleocurrent direction across central Iberia imply the presence of a second more southerly orocline, the Central Iberian Orocline, that is continuous with, but convex in the opposite direction of the Cantabrian Orocline. Together, the Cantabrian and Central Iberian oroclines define an S-shaped pair of continental-scale buckle folds. Palinspastic restoration of the oroclines yields a linear continental margin > 1500 km long characterized by consistent offshore paleoflow to the west, defining a westerly oceanic domain (presumably the Rheic Ocean) and an easterly landward direction (presumably Gondwana). Recognition of the southern orocline explains the unusual width of the orogen, the geometry of aeromagnetic anomalies attributable to Variscan rocks, and is consistent with available structural data, paleomagnetic declination data, and the distribution of correlative Paleozoic and older rock sequences including shallow water strata of the Gondwanan margin. The S-geometry of the oroclines is similar to the Z-geometry of the Alaskan oroclines of the North America Cordillera, suggesting that as in the Cordillera, the Iberian oroclines may record a significant, margin-parallel translation event.