time_filter.php

<?php

/**
 * @param array $reports
 * @param string $start
 * @param string $end
 * @return array
 */
function timeFilter(array $reports, $start, $end)
{
    $start = strtotime($start);
    $end = strtotime($end);

    return array_filter($reports, function(array $report) use ($start, $end) {
        $current = strtotime($report['date']) ;
        return $start <= $current && $current < $end;
    });
}

$reports = [
    ['id' => 1, 'date' => '2015-01-01 00:00:00'],
    ['id' => 2, 'date' => '2015-01-02 00:00:00'],
    ['id' => 3, 'date' => '2015-01-03 00:00:00'],
    ['id' => 4, 'date' => '2015-01-04 00:00:00'],
    ['id' => 5, 'date' => '2015-01-05 00:00:00'],
];

var_export(timeFilter($reports, '2015-01-02', '2015-01-05'));

# => array (
#   1 =>
#   array (
#     'id' => 2,
#     'date' => '2015-01-02 00:00:00',
#   ),
#   2 =>
#   array (
#     'id' => 3,
#     'date' => '2015-01-03 00:00:00',
#   ),
#   3 =>
#   array (
#     'id' => 4,
#     'date' => '2015-01-04 00:00:00',
#   ),
# )