The dissertation addresses direct adaptive control frameworks for Lyapunov stabilization of the MIMO nonlinear uncertain systems for both uncertain discrete-time and continuous-time systems. For system theory, the development of continuous-time theory always comes along with its discrete-time counterpart. However, for direct adaptive control frameworks we find relative few Lyapunov-based results published, which is mainly due to difficulty to find feasible Lyapunov candidates and to prove negative definiteness of the Lyapunov difference. Furthermore, digital computer is widely used in all fields. Most of time, we have to deal with the direct source of discrete-time signals, even the discrete-time signals arise from continuous-time settings as results of measurement or data collection process. These motivate our study in this field. For discrete-time systems, we have investigated the results with trajectory dependent hypothesis, where the Lyapunov candidate function V combines the information from the current state k and one step ahead k-1 along the track x(k), for k≥0. The proposed frameworks guarantee partial stability of the closed-loop systems, such that the feedback gains stabilize the closed-loop system without the knowledge of the system parameters. In addition, our results show that the adaptive feedback laws can be characterized by Kronecker calculus. Later, we release this trajectory dependent hypothesis for normal discrete-time nonlinear systems. At the same time, the continuous-time cases are also studied when system with matched disturbances, where the disturbances can be characterized by known continuous function matrix and unknown parameters. Here, the trajectory dependent Lyapunov candidates (tdLC), so long as the time step |t(k)-t(k-1) | ≤ £_ and the corresponding track |x(k)-x(k-1)| ≤ £` are sufficiently small, only exist in discrete-time case. In addition, we have extended the above control designs to systems with exogenous disturbances and £d2 disturbances. Finally, we develop a robust direct adaptive control framework for linear uncertain MIMO systems under the variance of unknow system matrix from given stable solution is bounded, that is |A-Ac| ¡Ý |B Kg| ≤ |£GA|. In general, through Lyapunov-based design we can obtain the global solutions and direct adaptive control design can simultaneously achieve parameter estimation and closed-loop stability.

It has been proposed that as animals explore their environment they build and maintain a cognitive map, an internal representation of their surroundings (Tolman, 1948). We tested this hypothesis using a task designed to assess the ability of rats to make a spatial inference (take a novel shortcut)(Roberts et al., 2007). Our findings suggest that rats are unable to make a spontaneous spatial inference. Furthermore, they bear similarities to experiments which have been similarly unable to replicate or support Tolman’s (1948) findings. An inability to take novel shortcuts suggests that rats do not possess a cognitive map (Bennett, 1996). However, we found evidence of alternative learning strategies, such as latent learning (Tolman & Honzik, 1930b) , which suggest that rats may still be building such a representation, although it does not appear they are able to utilise this information to make complex spatial computations. Neurons found in the hippocampus show remarkable spatial modulation of their firing rate and have been suggested as a possible neural substrate for a cognitive map (O'Keefe & Nadel, 1978). However, the firing of these place cells often appears to be modulated by features of an animal’s behaviour (Ainge, Tamosiunaite, et al., 2007; Wood, Dudchenko, Robitsek, & Eichenbaum, 2000). For instance, previous experiments have demonstrated that the firing rate of place fields in the start box of some mazes are predictive of the animal’s final destination (Ainge, Tamosiunaite, et al., 2007; Ferbinteanu & Shapiro, 2003). We sought to understand whether this prospective firing is in fact related to the goal the rat is planning to navigate to or the route the rat is planning to take. Our results provide strong evidence for the latter, suggesting that rats may not be aware of the location of specific goals and may not be aware of their environment in the form of a contiguous map. However, we also found behavioural evidence that rats are aware of specific goal locations, suggesting that place cells in the hippocampus may not be responsible for this representation and that it may reside elsewhere (Hok, Chah, Save, & Poucet, 2013). Unlike their typical activity in an open field, place cells often have multiple place fields in geometrically similar areas of a multicompartment environment (Derdikman et al., 2009; Spiers et al., 2013). For example, Spiers et al. (2013) found that in an environment composed of four parallel compartments, place cells often fired similarly in multiple compartments, despite the active movement of the rat between them. We were able to replicate this phenomenon, furthermore, we were also able to show that if the compartments are arranged in a radial configuration this repetitive firing does not occur as frequently. We suggest that this place field repetition is driven by inputs from Boundary Vector Cells (BVCs) in neighbouring brain regions which are in turn greatly modulated by inputs from the head direction system. This is supported by a novel BVC model of place cell firing which predicts our observed results accurately. If place cells form the neural basis of a cognitive map one would predict spatial learning to be difficult in an environment where repetitive firing is observed frequently (Spiers et al., 2013). We tested this hypothesis by training animals on an odour discrimination task in the maze environments described above. We found that rats trained in the parallel version of the task were significantly impaired when compared to the radial version. These results support the hypothesis that place cells form the neural basis of a cognitive map; in environments where it is difficult to discriminate compartments based on the firing of place cells, rats find it similarly difficult to discriminate these compartments as shown by their behaviour. The experiments reported here are discussed in terms of a cognitive map, the likelihood that such a construct exists and the possibility that place cells form the neural basis of such a representation. Although the results of our experiments could be interpreted as evidence that animals do not possess a cognitive map, ultimately they suggest that animals do have a cognitive map and that place cells form a more than adequate substrate for this representation.